Noncommutative generalizations of theorems of Cohen and Kaplansky

Noncommutative generalizations of
theorems of Cohen and Kaplansky

Manuel L. Reyes Department of Mathematics
University of California
Berkeley, CA, 94720-3840
Department of Mathematics
University of California, San Diego
9500 Gilman Drive, #0112
La Jolla, CA 92093-0112 m1reyes@math.ucsd.edu http://math.ucsd.edu/ m1reyes/
June 2, 2011
Abstract.

This paper investigates situations where a property of a ring can be tested on a set of “prime right ideals.” Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every “prime right ideal” is finitely generated (resp. principal), where the phrase “prime right ideal” can be interpreted in one of many different ways. We also use our methods to show that other properties can be tested on special sets of right ideals, such as the right artinian property and various homological properties. Applying these methods, we prove the following noncommutative generalization of a result of Kaplansky: a (left and right) noetherian ring is a principal right ideal ring iff all of its maximal right ideals are principal. A counterexample shows that the left noetherian hypothesis cannot be dropped. Finally, we compare our results to earlier generalizations of Cohen’s and Kaplansky’s theorems in the literature.

Key words and phrases:
point annihilators, cocritical right ideals, Cohen’s Theorem, right noetherian rings, Kaplansky’s Theorem, principal right ideals, Krull dimension, semiprime rings, essential right ideals
Key words and phrases:
point annihilators and cocritical right ideals and Cohen’s Theorem and right noetherian rings and Kaplansky’s Theorem and principal right ideals
2010 Mathematics Subject Classification:
Primary: 16D25, 16P40, 16P60; Secondary: 16N60

1. Introduction

Two famous theorems from commutative algebra due to I. S. Cohen and I. Kaplansky state that, to check whether every ideal in a commutative ring is finitely generated (respectively, principal), it suffices to test only the prime ideals. Cohen’s Theorem appeared as Theorem 2 of [6].

Theorem 1.1 (Cohen’s Theorem).

A commutative ring is noetherian iff every prime ideal of is finitely generated.

Also, we recall a characterization of commutative principal ideal rings due to I. Kaplansky, which appeared as Theorem 12.3 of [18]. Throughout this paper, a ring in which all right ideals are principal will be called a principal right ideal ring, or PRIR. Similarly, we have principal left ideal rings (PLIRs), and a ring which is both a PRIR and a PLIR is called a principal ideal ring, or PIR.

Theorem 1.2 (Kaplansky’s Theorem).

A commutative noetherian ring is a principal ideal ring iff every maximal ideal of is principal.

Combining this result with Cohen’s Theorem, Kaplansky deduced the following in Footnote 8 on p. 486 of [18].

Theorem 1.3 (Kaplansky-Cohen Theorem).

A commutative ring is a principal ideal ring iff every prime ideal of is principal.

(We refer to this result as the Kaplansky-Cohen Theorem for two reasons. The primary and most obvious reason is that it follows from a combination of the above results due to Cohen and Kaplansky. But we also use this term because it is a result in the spirit of Cohen’s Theorem, that was first deduced by Kaplansky.)

The unifying theme of this paper is the generalization of the above theorems to noncommutative rings, using certain families of right ideals as our tools. Let us mention a typical method of proof of these theorems, as this will guide our investigation into their noncommutative generalizations. One first assumes that the prime ideals of a commutative ring are all finitely generated (or principal), but that there exists an ideal of that is not f.g. (or principal). One then passes to a “maximal counterexample” by Zorn’s Lemma and proves that such a maximal counterexample is prime. This contradicts the assumption that all primes have the relevant property, proving the theorem.

These “maximal implies prime” theorems were systematically studied in [29] from the viewpoint of certain families of ideals, called Oka families. In particular, the Cohen and Kaplansky-Cohen theorems were recovered in [29, p. 3017]. These families were generalized to noncommutative rings in [33], where we defined Oka families of right ideals. This resulted in a noncommutative generalization of Cohen’s Theorem in [33, Thm. 6.2], stating that a ring is right noetherian iff its completely prime right ideals are all finitely generated. In the present paper, we will improve upon this result, providing smaller “test sets” of right ideals that need to be checked to determine if a ring is right noetherian. In addition, we will provide generalizations of Kaplansky’s Theorem 1.2 and the Kaplansky-Cohen Theorem 1.3.

We begin by reviewing the relevant results from [29] in §2. This includes an introduction to the notions of right Oka families, classes of cyclic modules closed under extensions, completely prime right ideals, the Completely Prime Ideal Principle (CPIP), and the CPIP Supplement. Our work in §§34 addresses the following question: what are some sufficient conditions for all right ideals of a ring to lie in a given right Oka family? In §3 we develop the idea of a (noetherian) point annihilator set in order to deal with this problem. Then in §4 we prove the Point Annihilator Set Theorem 4.1. Along with its consequences, such as Theorem 4.3, this theorem gives sufficient conditions for a property of right ideals to be testable on a smaller set of right ideals. We achieve a generalization of Cohen’s Theorem in Theorem 4.5. This result is “flexible” in the sense that, in order to check whether a ring is right noetherian, one can use various test sets of right ideals (in fact, certain point annihilator sets will work). One important such set is the cocritical right ideals. Other consequences of the Point Annihilator Set Theorem are also investigated.

Next we consider families of principal right ideals in §5. Whereas the family of principal ideals of a commutative ring is always an Oka family, it turns out that the family of principal right ideals can fail to be a right Oka family in certain noncommutative rings. By defining a right Oka family that “approximates” , we are able to provide a noncommutative generalization of the Kaplansky-Cohen Theorem in Theorem 5.11. As before, a specific version of this theorem is the following: a ring is a principal right ideal ring iff all of its cocritical right ideals are principal.

In §6 we sharpen our versions of the Cohen and Kaplansky-Cohen Theorems by considering families of right ideals that are closed under direct summands. This allows us to reduce the “test sets” of the Point Annihilator Set Theorem 4.3 to sets of essential right ideals. For instance, to check if a ring is right noetherian or a principal right ideal ring, it suffices to test the essential cocritical right ideals. Other applications involving homological properties of right ideals are considered.

We work toward a noncommutative generalization of Kaplansky’s Theorem 1.2 in §7. The main result here is Theorem 7.9, which states that a (left and right) noetherian ring is a principal right ideal ring iff its maximal right ideals are principal. Notably, our analysis also implies that such a ring has right Krull dimension . An example shows that the theorem does not hold if the left noetherian hypothesis is omitted.

Finally, we explore the connections between our results and previous generalizations of the Cohen and Kaplansky-Cohen theorems in §8. These results include theorems due to V. R. Chandran, K. Koh, G. O. Michler, P. F. Smith, and B. V. Zabavs’kiĭ. Discussing these earlier results affords us an opportunity to survey some previous notions of “prime right ideals” studied in the literature.

Conventions

All rings are associative and have identity, and all modules and ring homomorphisms are unital. Let be a ring. We say is a semisimple ring if is a semisimple module. We denote the Jacobson radical of by . We say that is semilocal (resp. local) if is semisimple (resp. a division ring). Given a family of right ideals in a ring , we let denote the complement of within the set of all right ideals of . Now fix an -module . We will write to mean that is an essential submodule of . A proper factor of is a module of the form for some nonzero submodule .

2. Review of right Oka families

In [33], we introduced the following notion of a “one-sided prime.”

Definition 2.1.

A right ideal is a completely prime right ideal if, for all ,

Notice immediately that a two-sided ideal is completely prime as a right ideal iff it is a completely prime ideal (that is, the factor ring is a domain). In particular, the completely prime right ideals of a commutative ring are precisely the prime ideals of that ring.

One way in which these right ideals behave like prime ideals of commutative rings is that right ideals that are maximal in certain senses tend to be completely prime. A more precise statement requires a definition. Given a right ideal and element of a ring , we denote

Definition 2.2.

A family of right ideals in a ring is an Oka family of right ideals (or a right Oka family) if and, for any element and any right ideal ,

For a family of right ideals in a ring , we let denote the complement of (the set of right ideals of that do not lie in ), and we let denote the set of right ideals of that are maximal in . The precise “maximal implies prime” result, which was proved in [33, Thm. 3.6], can now be stated.

Theorem 2.3 (Completely Prime Ideal Principle).

Let be an Oka family of right ideals in a ring . Then any right ideal is completely prime.

A result accompanying the Completely Prime Ideal Principle (CPIP) shows that, for special choices of right Oka families , in order to test whether consists of all right ideals it is enough to check that all completely prime right ideals lie in . Throughout this paper, a family of right ideals in a ring is called a semifilter if, whenever and is a right ideal of , implies .

Theorem 2.4 (Completely Prime Ideal Principle Supplement).

Let be a right Oka family in a ring such that every nonempty chain of right ideals in (with respect to inclusion) has an upper bound in . Let denote the set of completely prime right ideals of .

  • Let be a semifilter of right ideals in . If , then .

  • For , if all right ideals in containing (resp. properly containing ) belong to , then all right ideals containing (resp. properly containing ) belong to .

  • If , then consists of all right ideals of .

In order to efficiently construct right Oka families, we established the following correspondence in [33, Thm. 4.7]. A class of cyclic right -modules is said to be closed under extensions if and, for every short exact sequence of cyclic right -modules, and imply . Given a class of cyclic right -modules, one may construct the following family of right ideals of :

Conversely, given a family of right ideals in , we construct a class of cyclic -modules

Theorem 2.5.

Given a class of cyclic right -modules that is closed under extensions, the family is a right Oka family. Conversely, given a right Oka family , the class is closed under extensions.

This theorem was used to construct a number of examples of right Oka families in [33]. For us, the most important such example is the finitely generated right ideals: in any ring , the family of finitely generated right ideals of is a right Oka family (see [33, Prop. 3.7]).

An easy consequence of the above theorem, proved in [33, Cor. 4.9], will be useful throughout this paper.

Corollary 2.6.

Let be a right Oka family in a ring . Suppose that is such that the right -module has a filtration

where each filtration factor is cyclic and of the form for some . Then .

An important issue to be dealt with in the proof of Theorem 2.5 is that of similarity of right ideals. Two right ideals and of a ring are similar, written , if as right -modules. Two results from [33, §4] about isomorphic cyclic modules will be relevant to the present paper, the second of which deals directly with similarity of right ideals.

Lemma 2.7.

For any ring with right ideal and element , the following cyclic right -modules are isomorphic: .

Proposition 2.8.

Every right Oka family in a ring is closed under similarity: if then every right ideal similar to is also in .

Finally, in [33, §6] we studied a special collection of completely prime right ideals.

Definition 2.9.

A module is monoform if, for every nonzero submodule , every nonzero homomorphism is injective. A right ideal is comonoform if the right -module is monoform.

It was shown in [33, Prop. 6.3] that every comonoform right ideal is completely prime. The idea is that comonoform right ideals form an especially “well-behaved” subset of the completely prime right ideals of any ring.

3. Point annihilator sets for classes of modules

In this section we develop an appropriate notion of a “test set” for certain properties of right ideals in noncommutative rings. This is required for us to state the main theorems along these lines in the next section. Recall that a point annihilator of a module is defined to be an annihilator of a nonzero element .

Definitions 3.1.

Let be a class of right modules over a ring . A set of right ideals of is a point annihilator set for if every nonzero has a point annihilator that lies in . In addition, we make the following two definitions for special choices of :

  • A point annihilator set for the class of all right -modules will simply be called a (right) point annihilator set for .

  • A point annihilator set for the class of all noetherian right -modules will be called a (right) noetherian point annihilator set for .

Notice that a point annihilator set need not contain the unit ideal , because point annihilators are always proper right ideals. Another immediate observation is that, for a right noetherian ring , a right point annihilator set for is the same as a right noetherian point annihilator set for .

Remark 3.2.

The idea of a point annihilator set for a class of modules is simply that is “large enough” to contain a point annihilator of every nonzero module in . In particular, our definition does not require every right ideal in to actually be a point annihilator for some module in . This means that any other set of right ideals with is also a point annihilator set for . On the other hand, if is a subclass of modules, then is again a point annihilator set for .

Remark 3.3.

Notice that is a point annihilator set for a class of modules iff, for every nonzero module , there exists a proper right ideal such that the right module embeds into .

The next result shows that noetherian point annihilator sets for a ring exert a considerable amount of control over the noetherian right -modules.

Lemma 3.4.

A set of right ideals in is a noetherian point annihilator set iff for every noetherian module , there is a finite filtration of

such that, for , there exists such that .

Proof.

The “if” direction is easy, so we will prove the “only if” part. For convenience, we will refer to a filtration like the one described above as as an -filtration. Suppose that is a noetherian point annihilator set for , and let be noetherian. We prove by noetherian induction that has an -filtration. Consider the set of nonzero submodules of that have an -filtration. Because is a noetherian point annihilator set, it follows that is nonempty. Since is noetherian, has a maximal element, say . Assume for contradiction that . Then is noetherian, and by hypothesis there exists with such that for some . But then , contradicting the maximality of . Hence , completing the proof. ∎

We wish to highlight a special type of point annihilator set in the definition below.

Definition 3.5.

A set of right ideals of a ring is closed under point annihilators if, for all , every point annihilator of lies in . (This is equivalent to saying that and imply .) If is a class of right -modules, we will say that is a closed point annihilator set for if is a point annihilator set for and is closed under point annihilators.

The idea of the above definition is that is “closed under passing to further point annihilators of ” whenever . The significance of these closed point annihilator sets is demonstrated by the next result.

Lemma 3.6.

Let be a class of right modules over a ring that is closed under taking submodules (e.g. the class of noetherian modules). Suppose that is a closed point annihilator set for . Then for any other point annihilator set of , the set is a point annihilator set for .

Proof.

Let . Because is a point annihilator set for , there exists such that . By the hypothesis on , the module lies in . Because is also a point annihilator set for , there exists such that . The fact that is closed implies that . This proves that is also a point annihilator set for . ∎

The prototypical example of a noetherian point annihilator set is the prime spectrum of a commutative ring. In fact, every noetherian point annihilator set in a commutative ring can be “reduced to” some set of prime ideals, as we show below.

Proposition 3.7.

In any commutative ring , the set of prime ideals is a closed noetherian point annihilator set. Moreover, given any noetherian point annihilator set for , the set is a noetherian point annihilator subset of consisting of prime ideals.

Proof.

The set is a noetherian point annihilator set thanks to the standard fact that any noetherian module over a commutative ring has an associated prime; see, for example, [8, Thm. 3.1]. Furthermore, this set is closed because for , the annihilator of any nonzero element of is equal to . The last statement now follows from Lemma 3.6. ∎

In this sense right noetherian point annihilator sets of a ring generalize the concept of the prime spectrum of a commutative ring. However, one should not push this analogy too far: in a commutative ring , any set of ideals containing is also a noetherian point annihilator set! In fact, with the help of Proposition 3.7 it is easy to verify that any commutative ring has smallest noetherian point annihilator set , and that a set of ideals of is a noetherian point annihilator set for iff .

For most of the remainder of this section, we will record a number of examples of point annihilator sets that will be useful in later applications. Perhaps the easiest example is the following: the family of all right ideals of a ring is a point annihilator set for any class of right -modules. A less trivial example: the family of maximal right ideals of a ring is a point annihilator set for the class of right -modules of finite length, or for the larger class of artinian right modules. More specifically, according to Remark 3.3 it suffices to take any set of maximal right ideals such that the exhaust all isomorphism classes of simple right modules.

Example 3.8.

Recall that a module is said to be semi-artinian if every nonzero factor module of has nonzero socle, and that a ring is right semi-artinian if is a semi-artinian module. One can readily verify that is right semi-artinian iff every nonzero right -module has nonzero socle. Thus for such a ring , the set of maximal right ideals is a point annihilator set for , and in particular it is a noetherian point annihilator set for .

Example 3.9.

Let be a left perfect ring, that is, a semilocal ring whose Jacobson radical is left -nilpotent—see [25, §23] for details. (Notice that this class of rings includes semiprimary rings, especially right or left artinian rings.) By a theorem of Bass (see [25, (23.20)]), over such a ring, every right -module satisfies DCC on cyclic submodules. Thus every nonzero right module has nonzero socle, and such a ring is right semi-artinian. But has finitely many simple modules up to isomorphism (because the same is true modulo its Jacobson radical). Choosing a set of maximal right ideals such that the modules exhaust the isomorphism classes of simple right -modules, we conclude by Remark 3.3 that is a point annihilator set for any class of right modules . Hence forms a right noetherian point annihilator set for . (The observant reader will likely have noticed that the same argument applies more generally to any right semi-artinian ring with finitely many isomorphism classes of simple right modules.)

Directly generalizing the fact that the prime spectrum of a commutative ring is a noetherian point annihilator set, we have the following fact, valid for any noncommutative ring.

Proposition 3.10.

The set of completely prime right ideals in any ring is a noetherian point annihilator set.

Proof.

Let be noetherian. For any point annihilator with , the module is noetherian. Thus must have a maximal point annihilator , and is completely prime by [33, Prop. 5.3]. ∎

Recall that in any ring , the set of comonoform right ideals of forms a subset of the set of all completely prime right ideals of . As we show next, the subset of comonoform right ideals is also a noetherian point annihilator set.

Proposition 3.11.

For any ring , the set of comonoform right ideals in is a closed noetherian point annihilator set.

Proof.

Because a nonzero submodule of a monoform module is again monoform, Remark 3.3 shows that it is enough to check that any nonzero noetherian module has a monoform submodule. This has already been noted, for example, in [30, 4.6.5]. We include a separate proof for the sake of completeness.

Let be maximal with respect to the property that there exists a nonzero cyclic submodule that can be embedded in . It is readily verified that is monoform, and writing for some comonoform right ideal , the fact that embeds in shows that is a point annihilator of . ∎

Our most “refined” instance of a noetherian point annihilator set for a general noncommutative ring is connected to the concept of (Gabriel-Rentschler) Krull dimension. We review the relevant definitions here, and we refer the reader to the monograph [14] or the textbooks [13, Ch. 15] or [30, Ch. 6] for further details. Define by induction classes of right -modules for each ordinal (for convenience, we consider to be an ordinal number) as follows. Set to be the class consisting of the zero module. Then for an ordinal such that has been defined for all ordinals , define to be the class of all modules such that, for every descending chain

of submodules of indexed by natural numbers, one has for almost all indices . Now if a module belongs to some , its Krull dimension, denoted , is defined to be the least ordinal such that . Otherwise we say that the Krull dimension of does not exist.

From the definitions it is easy to see that the right -modules of Krull dimension 0 are precisely the (nonzero) artinian modules. Also, a module has Krull dimension 1 iff it is not artinian and in every descending chain of submodules of , almost all filtration factors are artinian.

One of the more useful features of the Krull dimension function is that it is an exact dimension function, in the sense that, given an exact sequence of right -modules, one has

where either side of the equation exists iff the other side exists. See [13, Lem. 15.1] or [30, Lem. 6.2.4] for details.

The Krull dimension can also be used as a dimension measure for rings. We define the right Krull dimension of a ring to be . The left Krull dimension of is defined similarly.

Now a module is said to be -critical ( an ordinal) if but for all , and we say that is critical if it is -critical for some ordinal . With this notion in place, we define a right ideal to be -cocritical if the module is -critical, and we say that is cocritical if it is -cocritical for some ordinal . Notice immediately that a 0-critical module is the same as a simple module, and the 0-cocritical right ideals are precisely the maximal right ideals.

Cocritical right ideals were already studied by A. W. Goldie in [11], though they are referred to there as “critical” right ideals. (The reader should take care not to confuse this terminology with the phrase “critical right ideal” used in a different sense elsewhere in the literature, as mentioned in [33, §6].)

Remarks 3.12.

The first two remarks below are known; for example, see [30, §6.2].

  • Every nonzero submodule of a critical module is also critical and has . Suppose that is -critical. If , then because , the exactness of Krull dimension would imply the contradiction . Hence . Also, for any nonzero submodule we have , proving that is -critical.

  • A critical module is always monoform. Suppose that is -critical and fix a nonzero homomorphism where . Because and are both nonzero submodules of , they are also -critical by (1). Then , so we must have . Thus is indeed monoform.

  • Any cocritical right ideal is comonoform and, in particular, is completely prime. This follows immediately from the preceding remark and the fact [33, Prop. 6.3] that any comonoform right ideal is completely prime.

It is possible to characterize the (two-sided) ideals that are cocritical as right ideals.

Proposition 3.13.

For any ring and any ideal , the following are equivalent:

  • is cocritical as a right ideal;

  • is a (right Ore) domain with right Krull dimension.

Proof.

Because every cocritical right ideal is comonoform, (1)(2) follows from the fact [33, Prop. 6.5] that a two-sided ideal is comonoform as a right ideal iff its factor ring is a right Ore domain. Because a semiprime ring with right Krull dimension is right Goldie, the two conditions in (2) are equivalent.

(2)(1): It can be shown that, for every module whose Krull dimension exists and for every injective endomorphism , one has (see [13, Lem. 15.6]). Applying this to , we see that for all nonzero (this is also proved in [30, Lem. 6.3.9]). Thus , and consequently , are critical modules. ∎

Example 3.14.

The last proposition is useful for constructing an ideal of a ring that is (right and left) comonoform but not (right or left) cocritical. If is a commutative domain that does not have Krull dimension, then the zero ideal of is prime and thus is comonoform by [33, Prop. 6.5]. But because does not have Krull dimension, the zero ideal cannot be cocritical by the previous result. For an explicit example, one can take for some commutative domain . It is shown in [14, Ex. 10.1] that such a ring does not have Krull dimension, using the fact that a polynomial ring has right Krull dimension iff the ground ring is right noetherian.

The reason for our interest in the set of cocritical right ideals is that it is an important example of a noetherian point annihilator set in a general ring.

Proposition 3.15.

For any ring , the set of all cocritical right ideals is a closed point annihilator set for the class of right -modules whose Krull dimension exists. In particular, this set is a closed noetherian point annihilator set for .

Proof.

Because any nonzero module with Krull dimension has a critical submodule (see [13, Lem. 15.8] or [30, Lem. 6.2.10]), Remark 3.3 shows that the set of cocritical right ideals of is a point annihilator set for the class of right -modules with Krull dimension. Because any noetherian module has Krull dimension (see [13, Lem. 15.3] or [30, Lem. 6.2.3]), we see by Remark 3.2 that this same set is a right noetherian point annihilator set for . The fact that this set is closed under point annihilators follows from Remark 3.12(1). ∎

Let us further examine the relationship between the general noetherian point annihilator sets given in Propositions 3.103.11, and 3.15. From [33, Prop. 6.3] and Remark 3.12(3) we see that there are always the following containment relations (where the first three sets are noetherian point annihilator sets but the last one is not, in general):

(3.1)

Notice that in a commutative ring the first two sets are equal to by [33, Cor. 2.3 & Cor. 6.7], and when is commutative and has Gabriel-Rentschler Krull dimension (e.g., when is noetherian) the third set is also equal to by Proposition 3.13. The latter fact provides many examples where the last containment is strict: in any commutative ring with Krull dimension there exists a nonmaximal prime ideal, which must be cocritical. It was shown in [33, Lem. 6.4 ff.] and Example 3.14 that the first two inclusions can each be strict. However, the latter example was necessarily non-noetherian. Below we give an example of a noncommutative artinian (hence noetherian) ring over which both containments are strict. This example makes use of the following characterization of semi-artinian monoform modules. The proof is straightforward and therefore is omitted. The socle of a module is denoted by .

Lemma 3.16.

Let be a semi-artinian -module. Then the following are equivalent:

  • is monoform;

  • For any nonzero submodule , and do not have isomorphic nonzero submodules;

  • is simple and does not embed into any proper factor module of .

Example 3.17.

Let be a division ring, and let be the ring of all matrices over of the form

(3.2)

One can easily verify (for example, by passing to the factor of by its Jacobson radical) that has two simple right modules up to isomorphism. We may view these modules as with right -action given by right multiplication by in (3.2) and with right action given by right multiplication by in (3.2). Consider the right ideals

Then the cyclic module is isomorphic to the space of row vectors with the natural right -action. Notice that () corresponds to the submodule of row vectors whose first entries are zero. One can check that the only submodules of are , which implies that this is the unique composition series of . It is clear that

We claim that is a completely prime right ideal that is not comonoform. To see that it is completely prime, it suffices to show that every nonzero endomorphism of is injective. Indeed, the only proper factors of are and . By an inspection of composition factors, neither of these can embed into , proving that is completely prime. To see that is not comonoform, consider that

By Lemma 3.16 we see that is not monoform and thus is not comonoform.

We also claim that is comonoform but not cocritical. Notice that over any right artinian ring, every cyclic critical module has Krull dimension 0. But a 0-critical module is necessarily simple. Thus a cocritical right ideal in a right artinian ring must be maximal. But is not maximal and thus is not cocritical. On the other hand, has unique composition series . This allows us to easily verify, using Lemma 3.16, that is monoform, proving that is comonoform.

This same example also demonstrates that the set of completely prime right ideals is not always closed under point annihilators (as in Definition 3.5). This is because the cyclic submodule certainly has a nonzero noninjective endomorphism, as both of its composition factors are isomorphic.

An example along these lines was already used in [14, p. 11] to show that a monoform module need not be critical. Notice that the completely prime right ideal above is such that is uniform, even if it is not monoform. (This means that the right ideal is “meet-irreducible.”) An example of a completely prime right ideal whose factor module is not uniform was already given in Example completely prime not meet-irreducible.

Given the containments of noetherian point annihilator sets in (3.1), one might question the need for the notion of a point annihilator set. Why not simply state all theorems below just for the family of cocritical right ideals? We already have an answer to this question in Example 3.9, which demonstrates that every left perfect ring has a finite right noetherian point annihilator set. The reason we can reduce to a finite set in such rings is the fact stated in Remark 3.3 that a noetherian right module only needs to contain a submodule isomorphic to for some . In other words, only needs to contain a single representative from any given similarity class. So while a left perfect ring may have infinitely many maximal right ideals, it has only finitely many similarity classes of maximal right ideals. Thus we can reduce certain problems about all right ideals of to a finite set of maximal ideals! This will be demonstrated in Proposition 4.8 and Corollary 5.5, below where we shall prove that a left perfect ring is right noetherian (resp. a PRIR) iff all maximal right ideals belonging to a (properly chosen) finite set are finitely generated (resp. principal).

We have also phrased the discussion in terms of general noetherian point annihilator sets to leave open the possibility of future applications to classes of rings which have nicer noetherian point annihilator sets than the whole set of cocritical right ideals, akin to the class of left perfect rings.

4. The Point Annihilator Set Theorem

Having introduced the notion of a point annihilator set, we can now state our fundamental result, the Point Annihilator Set Theorem 4.1. This theorem gives conditions under which one may deduce that one family of right ideals is contained in a second family of right ideals. We will most often use it as a sufficient condition for concluding that all right ideals of a ring lie in a particular right Oka family .

Certain results in commutative algebra state that when every prime ideal in a commutative ring has a certain property, then all ideals in the ring have that property. As mentioned in the introduction, the two motivating examples are Cohen’s Theorem 1.1 and Kaplansky’s Theorem 1.3. In [29, p. 3017], these theorems were both recovered in the context of Oka families and the Prime Ideal Principle. The useful tool in that context was the “Prime Ideal Principle Supplement” [29, Thm. 2.6]. We have already provided one noncommutative generalization of this tool in the Theorem 2.4, which we used to produce a noncommutative extension of Cohen’s Theorem in [33, Thm. 3.8] stating that a ring is right noetherian iff its completely prime right ideals are finitely generated.

The CPIP Supplement states that for certain right Oka families , if the set of completely prime right ideals lies in , then all right ideals lie in . The main goal of this section is to improve upon this result by allowing the set to be any point annihilator set. This is achieved in Theorem 4.3 as an application of the Point Annihilator Set Theorem.

The Point Annihilator Set Theorem basically formalizes a general “strategy of proof.” For the sake of clarity, we present an informal sketch of this proof strategy before stating the theorem. Suppose that we want to prove that every module with the property also has the property . Assume for contradiction that there is a counterexample. Use Zorn’s Lemma to pass to a counterexample satisfying that is “critical” with respect to not satisfying , in the sense that every proper factor module of satisfies but itself does not satisfy . Argue that has a nonzero submodule that satisfies . Finally, use the fact that and have to deduce the contradiction that has .

Our theorem applies in the specific case where one’s attention is restricted to cyclic modules. In the outline above, we may think of the properties and to be, respectively, “ where ” and “ where .”

Theorem 4.1 (The Point Annihilator Set Theorem).

Let be a right Oka family such that every nonempty chain of right ideals in (with respect to inclusion) has an upper bound in .

  • Let be a semifilter of right ideals in . If is a point annihilator set for the class of modules , then .

  • For any right ideal , if is a point annihilator set for the class of modules such that and (resp. ), then all right ideals containing (resp. properly containing) belong to .

  • If is a point annihilator set for the class of modules , then consists of all right ideals of .

Proof.

Suppose that the hypotheses of (1) hold, and assume for contradiction that there exists . The assumptions on  allow us to apply Zorn’s Lemma to find with . Then because is a semifilter. The point annihilator hypothesis implies that there is a nonzero element such that . On the other hand, implies that . By maximality of , this means that . Because is a right Oka family, we arrive at the contradiction .

Parts (2) and (3) follows from (1) by taking to be, respectively, the set of all right ideals of (properly) containing or the set of all right ideals of . ∎

Notice that part (1) above remains true if we weaken the condition on chains in to the following: every nonempty chain in has an upper bound in . The latter condition holds if every is such that is a noetherian module, or more generally if satisfies the ascending chain condition (as a partially ordered set with respect to inclusion). However, we shall not make use this observation in the present work.

The following is an illustration of how Theorem 4.1 can be applied in practice. It is well-known that every finitely generated artinian module over a commutative ring has finite length. However, there exist finitely generated (even cyclic) artinian right modules over noncommutative rings that do not have finite length; for instance, see [26, Ex. 4.28]. Here we provide a sufficient condition for all finitely generated artinian right modules over a ring to have finite length.

Proposition 4.2.

If all maximal right ideals of a ring are finitely generated, then every finitely generated artinian right -module has finite length.

Proof.

It suffices to show that every cyclic artinian right -module has finite length. Let be the semifilter of right ideals such that is right artinian, and let be the right Oka family of right ideals such that has finite length. Our goal is then to show that . Because every nonzero cyclic artinian module has a simple submodule, we see that is a point annihilator set for the class . To apply Theorem 4.1(1) we will show that every nonempty chain in has an upper bound in . For this, it is enough to check that consists of finitely generated right ideals. The hypothesis implies that all simple right -modules are finitely presented. If then , being a repeated extension of finitely many simple modules, is finitely presented. It follows that is finitely generated. (The details of the argument that consists of f.g. right ideals are in [33, Cor 4.9].) ∎

In light of the result above, it would be interesting to find a characterization of the rings over which every finitely generated artinian right -module has finite length. How would such a characterization unite both commutative rings and the rings in which every maximal right ideal is finitely generated?

For our purposes, it will often best to use a variant of the theorem above. This variant keeps with the theme of Cohen’s and Kaplansky’s results (Theorems 1.11.3) of “testing” a property on special sets of right ideals.

Theorem 4.3.

Let be a right Oka family such that every nonempty chain of right ideals in (with respect to inclusion) has an upper bound in . Let be a set of right ideals that is a point annihilator set for the class of modules .

  • Let be a divisible semifilter of right ideals in . If , then .

  • For any ideal , if all right ideals in that contain belong to , then every right ideal containing belongs to .

  • If , then all right ideals of belong to .

Proof.

As in the previous result, parts (2) and (3) are special cases of part (1). To prove (1), Theorem 4.1 implies that it is enough to show that is a point annihilator set for the class of modules . Fixing such , the hypothesis of part (1) ensures that has a point annihilator in , say for some . Because and is divisible, the fact that implies that . Thus , providing a point annihilator of that lies in . ∎

We also record a version of Theorem 4.3 adapted especially for families of finitely generated right ideals. Because of its easier formulation, it will allow for simpler proofs as we provide applications of Theorem 4.3.

Corollary 4.4.

Let be a right Oka family in a ring that consists of finitely generated right ideals. Let be a noetherian point annihilator set for . Then the following are equivalent:

  • consists of all right ideals of ;

  • is a noetherian point annihilator set;

  • .

Proof.

Given any , any nonzero submodule of is the image of a right ideal properly containing , which must be finitely generated; thus is a noetherian right -module. Stated another way, the class consists of noetherian modules. Thus (1)(2) follows from Theorem 4.1(3) and (1)(3) follows from Theorem 4.3(3). ∎

As our first application of the simplified corollary above, we will finally present our noncommutative generalization of Cohen’s Theorem 1.1, improving upon [33, Thm. 3.8].

Theorem 4.5 (A noncommutative Cohen’s Theorem).

Let be a ring with a right noetherian point annihilator set . The following are equivalent:

  • is right noetherian;

  • Every right ideal in is finitely generated;

  • Every nonzero noetherian right -module has a finitely generated point annihilator;

  • Every nonzero noetherian right -module has a nonzero cyclic finitely presented submodule.

In particular, is right noetherian iff every cocritical right ideal is finitely generated.

Proof.

The family of finitely generated right ideals is a right Oka family by [33, Prop. 3.7]. The equivalence of (1), (2), and (3) thus follows directly from Corollary 4.4. Also, (3)(4) comes from the observation that a right ideal is a point annihilator of a module iff there is an injective module homomorphism , as well as the fact that is a finitely presented module iff is a finitely generated right ideal [24, (4.26)(b)]. The last statement follows from Proposition 3.15. ∎

In particular, if we take the set above to be the completely prime right ideals of , we recover [33, Thm. 3.8]. Our version of Cohen’s Theorem will be compared and contrasted with earlier such generalizations in §8.

The result above suggests that one might wish to drop the word “cyclic” in characterization (4). This is indeed possible. We present this as a separate result since it does not take advantage of the “formalized proof method” given in Theorem 4.1. However, this result does follow the informal “strategy of proof” outlined at the beginning of this section.

Proposition 4.6.

For a ring , the following are equivalent:

  • is right noetherian;

  • Every nonzero noetherian right -module has a nonzero finitely presented submodule.

Proof.

Using the numbering from Theorem 4.5, we have (1)(4)(5). Suppose that (5) holds, and assume for contradiction that there exists a right ideal of that is not finitely generated. Using Zorn’s Lemma, pass to that is maximal with respect to not being finitely generated. Then because every right ideal properly containing is f.g., the module is noetherian. By hypothesis, there is a finitely presented submodule . Then implies that is finitely generated, so that is finitely presented. Because is an extension of the two finitely presented modules and , is finitely presented [28, Ex. 4.8(2)]. But if is finitely presented then is finitely generated [24, (4.26)(b)]. This is a contradiction. ∎

The Akizuki-Cohen Theorem of commutative algebra (cf. [6, pp. 27–28]) states that a commutative ring is artinian iff it is noetherian and every prime ideal is maximal. Recall that a module is finitely cogenerated if any family of submodules of whose intersection is zero has a finite subfamily whose intersection is zero. In [29, (5.17)] consideration of the class of finitely cogenerated right modules led to the following “artinian version” of Cohen’s theorem: a commutative ring is artinian iff for all , is finitely generated and is finitely cogenerated. Here we generalize both of these results to the noncommutative setting.

Proposition 4.7.

For a ring with right noetherian point annihilator set , the following are equivalent:

  • is right artinian;

  • is right noetherian and for all , has finite length;

  • For all , is finitely generated and has finite length;

  • For all , is finitely generated and is finitely cogenerated;

  • is right noetherian and every cocritical right ideal of is maximal;

  • Every cocritical right ideal of is finitely generated and maximal.

Proof.

(1)(2)(3): It is well-known that is right artinian iff has finite length. This equivalence then follows from Corollary 4.4, Theorem 4.5, and the fact that is a right Oka family (see [33, Ex. 5.18(4)]).

(1)(4): It is known that a module is artinian iff every quotient of is finitely cogenerated (see [28, Ex. 19.0]). Because is a right Oka family (see [33, Ex. 5.18(1B)]), (1)(4) follows from Corollary 4.4.

We get (1)(5)(6) by applying the equivalence of (1), (2), and (3) to the case where is the set of cocritical right ideals of , noting that every artinian critical module is necessarily simple. ∎

Of course, the fact that a right noetherian ring is right artinian iff all of its cocritical right ideals are maximal follows from a direct argument involving Krull dimensions of modules. Indeed, given a right noetherian ring with right Krull dimension , choose a right ideal maximal with respect to . Then for any right ideal , ; hence is cocritical. So

The result now follows once we recall that the 0-critical modules are precisely the simple modules.

We also mention another noncommutative generalization of the Akizuki-Cohen Theorem due to A. Kertész, which states that a ring is right artinian iff it is right noetherian and for every prime ideal , is right artinian [19]. (We thank the referee for bringing this reference to our attention.)

Another application of Theorem 4.5 tells us when a right semi-artinian ring, especially a left artinian ring, is right artinian. (The definition of a right semi-artinian ring was recalled in Example 3.8.)

Proposition 4.8.

(1) A right semi-artinian ring is right artinian iff every maximal right ideal of is finitely generated.

(2) Let be a left perfect ring (e.g. a semiprimary ring, such as a left artinian ring) and let be maximal right ideals such that exhaust all isomorphism classes of simple right modules. Then is right artinian iff all of the are finitely generated.

Proof.

It is easy to check that a right semi-artinian ring is right artinian iff it is right noetherian. The proposition then follows from Theorem 4.5 and Examples 3.83.9. ∎

A result of B. Osofsky [32, Lem. 11] states that a left or right perfect ring with Jacobson radical is right artinian iff is finitely generated as a right -module. This applies, in particular, to left artinian rings. D. V. Huynh characterized which (possibly nonunital) left artinian rings are right artinian in [15, Thm. 1]. In the unital case, his characterization recovers Osofsky’s result above for the special class of left artinian rings. We can use our previous result to recover a weaker version of Osofsky’s theorem that implies Huynh’s result for unital left artinian rings.

Corollary 4.9.

Let be a ring with . The the following are equivalent:

  • is right artinian;

  • is left perfect and is a finitely generated right ideal;

  • is perfect and is a finitely generated right -module.

In particular, if is semiprimary (for instance, if it is left artinian), then is right artinian iff is finitely generated on the right.

Proof.

Because any right artinian ring is both perfect and right noetherian, we have (1)(3). For (3)(2), suppose that is perfect and that is right finitely generated. Then for some finitely generated submodule , . Since is right perfect, is right T-nilpotent. Then by “Nakayama’s Lemma” for right T-nilpotent ideals (see [25, (23.16)]) implies that is finitely generated.

Finally we show (2)(1). Suppose that is left perfect and that is finitely generated. For any maximal right ideal of , we have . Now is a right ideal of the semisimple ring and is therefore finitely generated. Because is also finitely generated, we see that itself is finitely generated. Since this is true for all maximal right ideals of , Proposition 4.8(2) implies that is right artinian. ∎

Next we give a condition for every finitely generated right module over a ring to have a finite free resolution (FFR). Notice that such a ring is necessarily right noetherian. Indeed, any module with an FFR is necessarily finitely presented. Thus if every f.g. right -module has an FFR, then for every right ideal the module must have an FFR and therefore must be finitely presented. It follows (from Schanuel’s Lemma [24, (5.1)]) that is finitely generated, and is right noetherian.

Proposition 4.10.

Let be a right noetherian point annihilator set for a ring (e.g. the set of cocritical right ideals). Then the following are equivalent.

  • Every finitely generated right -module has a finite free resolution;

  • For all , has a finite free resolution;

  • Every right ideal in has a finite free resolution.

Proof.

(1)(3): As mentioned before the proposition, if every f.g. right -module has a finite free resolution then is right noetherian. So every right ideal is finitely generated and therefore has a finite free resolution.

Next, (3)(2) follows from the easy fact that, given , if has a finite free resolution then so does . For (2)(1), let be the family of right ideals such that has a finite free resolution and assume that . This is a right Oka family according to [33, Ex. 5.12(5)]. Moreover, if then is finitely presented. As noted earlier, this implies that must be finitely generated [24, (4.26)(b)]. It follows from Corollary 4.4 that every right ideal of lies in . Because any finitely generated right -module is an extension of cyclic modules and because the property of having an FFR is preserved by extensions, we conclude that (1) holds. ∎

5. Families of principal right ideals

We will use to denote the family of principal right ideals of a ring . If the ring is understood from the context, we may simply use to denote this family.

A theorem of Kaplansky [18, Thm. 12.3 & Footnote 8] states that a commutative ring is a principal ideal ring iff its prime ideals are all principal. In [29, (3.17)] this theorem was recovered via the “PIP supplement.” It is therefore reasonable to hope that the methods presented here will lead to a generalization of this result. Specifically, we would like to know whether a ring is a principal right ideal ring (PRIR) if, say, every cocritical right ideal is principal. It turns out that this is in fact true, but the path to proving the result is not as straightforward as one might imagine. The obvious starting point is to ask whether the family of principal right ideals in an arbitrary ring is a right Oka family. Suppose that is a ring such that is a right Oka family. Then Corollary 4.4 readily applies to . However, it is not immediately clear whether or not is necessarily right Oka for every ring . The following proposition provides some guidance in this matter.

Proposition 5.1.

Let be a multiplicative set. Then is a right Oka family iff it is closed under similarity. In particular, for any ring , the family of principal right ideals is a right Oka family iff it is closed under similarity.

Proof.

Any right Oka family is closed under similarity by Proposition refOka is similarity closed. Conversely, assume that the family in question is closed under similarity. Suppose that , and write for some . In the short exact sequence of right -modules

observe that . Because is closed under similarity and , we must also have . Fix such that . Then because we have , and implies that . ∎

In particular, we have the following “first approximation” to our desired theorem.

Corollary 5.2.

Let be a right noetherian point annihilator set for . The following are equivalent:

  • is a principal right ideal ring;

  • is closed under similarity and every right ideal in is principal;

  • is closed under similarity and is a right noetherian point annihilator set.

Proof.

If is a PRIR, then is equal to the family of all right ideals in and therefore is closed under similarity. Also, by Proposition 5.1, if is closed under similarity then it is a right Oka family. These observations along with Corollary 4.4 establish the equivalence of (1)–(3). ∎

This provides some motivation to explore for which rings the family is closed under similarity (and consequently is a right Oka family). Recall that a ring is called right duo if every right ideal of is a two-sided ideal. It is easy to see that in any right duo ring, and particularly in any commutative ring, every family of right ideals is closed under similarity. This is because in such a ring , any right ideal is necessarily a two-sided ideal, so that can be recovered from the isomorphism class of . Thus Proposition 5.1 applies to show that is a right Oka family whenever is a right duo ring, such as a commutative ring. For commutative rings , the fact that is an Oka family was already noted in [29, (3.17)].

Another collection of rings in which is closed under similarity is the class of local rings. To show that this is the case, we use the fact [33, Prop. 4.6] that a family of right ideals of a ring is closed under similarity iff, for every element and right ideal of , and imply . Suppose that is local, and that and are such that and is principal. We want to conclude that is principal. Write for some and . Let denote the group of units of . If , then is principal. Else implies that and hence ( local implies that right invertible elements are invertible). But then , so that is principal as desired.

Remark 5.3.

In any ring , let be right ideals such that is principal and . Then is generated by at most two elements. To see this, apply Schanuel’s Lemma (for instance, see [24, (5.1)]) to the exact sequences

to get . The latter module is generated by at most two elements. Therefore , being isomorphic to a direct summand of this module, is generated by at most two elements. Thus we see that such is “not too far” from being principal. (Of course, the same argument shows that if is generated by at most elements and if is similar to , then is generated by at most elements.)

The analysis above also provides the following useful fact: if the module is cancellable in the category of (finitely generated) right -modules (or even in the category of finite direct sums of f.g. right ideals), then the family is closed under similarity (and hence is a right Oka family). Indeed, if this is the case, suppose that for right ideals and with principal. By the remark above, we have finitely generated and . With the assumption on we would have principal, proving to be closed under similarity. (In fact one can similarly show that, over such rings, the minimal number of generators of a f.g. right ideal is an invariant of the similarity class of .)

This provides another class of rings for which is a right Oka family, as follows. Recall that a ring is said to have (right) stable range 1 if, for , implies that for some (see [27, §1] for details). In [9, Thm. 2] E. G. Evans showed that for any ring with stable range 1, is cancellable in the full module category . Thus for any ring with stable range 1, is a right Oka family. The class of rings with stable range 1 includes all semilocal rings (see [25, (20.9)] or [27, (2.10)]), so that this generalizes the case of local rings discussed above.

A similar argument applies in the class of 2-firs. A ring is said to be a -fir (where “fir” stands for “free ideal ring”) if the free right -module of rank 2 has invariant basis number and every right ideal of generated by at most two elements is free. We claim that is closed under similarity if is a 2-fir. Suppose that is similar to a principal right ideal . As before, we have , and is generated by at most two elements. So , and where because is princpal. Thus with , and the invariant basis number of the latter free module implies that . Hence is a principal right ideal.

There is yet another way in which can be closed under similarity. Suppose that every finitely generated right ideal of is principal; rings satisfying this property are often called right Bèzout rings. Then is equal to the set of all f.g. right ideals of , which is a right Oka family by [33, Prop. 3.7]. A familiar class of examples of such rings is the class of von Neumann regular rings; in such rings, every finitely generated right ideal is a direct summand of , and therefore is principal.

We present a summary of the examples above.

Examples 5.4.

In each of the following types of rings, the family is closed under similarity and thus is a right Oka family:

  • Right duo rings (including commutative rings);

  • Rings with stable range 1 (including semilocal rings);

  • 2-firs;

  • Right Bèzout rings (including von Neumann regular rings).

One collection of semilocal rings that we have already mentioned is the class of left perfect rings. An application of Corollary 5.2 in this case gives the following.

Corollary 5.5.

Let be a left perfect ring (e.g. a semiprimary ring, such as a one-sided artinian ring), and let be maximal right ideals such that the represent all isomorphism classes of simple right -modules. Then is a PRIR iff all of the are principal right ideals.

Proof.

By Example 5.4(2), is an Oka family of right ideals in . By Example 3.9, the set is a right noetherian point annihilator set. The claim then follows from Corollary 5.2. ∎

As it turns out, the family can indeed fail to be right Oka, even in a noetherian domain! This will be shown in Example 5.7 below, with the help of the following lemma.

Lemma 5.6.

Let be a ring with an element that is not a left zero-divisor.

  • If and are right ideals of with , then

  • For any ,

Proof.

(A) The containment “” holds without any assumptions on , , or because . To show “” let , so that there exist and such that . Because , there exists such that ; notice that . Then we have for . Now , and because is not a left zero-divisor we have .

(B) Setting and , one may compute that and (using the fact that is not a left zero divisor). The claim follows directly from part (A). ∎

Example 5.7.

A ring in which is not a right Oka family. Let be a field and let be the first Weyl algebra over . Then is known to be a noetherian domain (which is simple if has characteristic 0). Define the right ideal

which is shown to be nonprincipal in [30, 7.11.8]. Because contains both and , we must have .

Because , Lemma 5.6(B) above (with ) implies that . Therefore we have and both members of with proving that is not a right Oka family. In fact we have where is not principal (the isomorphism follows from Lemma 2.7), showing explicitly that is not closed under similarity as predicted by Proposition 5.1. In agreement with Remark 5.3, is generated by two elements.

Notice that , where acts by right multiplication and acts as . If has characteristic 0 then this module is evidently simple, and because we see that is a maximal right ideal. If instead , then is evidently not simple, and not even artinian (the submodules form a strictly descending chain for ). But every proper factor of this module has finite dimension over and is therefore artinian. So we see that is 1-critical, making a 1-cocritical right ideal. Thus regardless of the characteristic of , the nonprincipal right ideal is cocritical.

On the other hand, when the ring is known to be a principal (right and left) ideal ring—see [30, 7.11.7]. Then is equal to the set of all right ideals in and thus is a right Oka family. So we see that the property “ is a right Oka family” is not Morita invariant.∎

It would be very desirable to eliminate the condition in Corollary 5.2 that is closed under similarity. It turns out that a suitable strengthening of the hypothesis on the point annihilator set will in fact allow us to discard that assumption. The following constructions will help us achieve this goal in Theorem 5.11 below. Recall that for right ideals and of a ring , we write to mean that and are similar.

Definition 5.8.

For any ring , we define

Alternatively, is the largest subset of that is closed under similarity.

As with