Relative polynomial closure and monadically Krull monoids of integer-valued polynomials
Let be a Krull domain and the ring of integer-valued polynomials on . For any , we explicitly construct a divisor homomorphism from , the divisor-closed submonoid of generated by , to a finite sum of copies of . This implies that is a Krull monoid.
For a discrete valuation domain, we give explicit divisor theories of various submonoids of . In the process, we modify the concept of polynomial closure in such a way that every subset of has a finite polynomially dense subset. The results generalize to , the ring of integer-valued polynomials on a subset, provided doesn’t have isolated points in -adic topology.
Key words and phrases:monoid, factorization, monadically Krull, divisor homomorphism, divisor theory, integer-valued polynomial, polynomial closure
2000 Mathematics Subject Classification:Primary 13F20; Secondary 20M13, 13A05, 13B25, 11C08, 11R09
The ring of integer-valued polynomials enjoys quite chaotic non-unique factorization, in the following sense: given any finite list of natural numbers , one can find a polynomial that has exactly essentially different factorizations into irreducible elements of , namely, one with irreducible factors, one with , etc. . In contrast to this, A. Reinhart  has shown that is monadically Krull for any unique factorization domain , which means that the divisor-closed submonoid generated by any single polynomial is a Krull monoid. So we have here an interesting case of Krull monoids with rather wild factorization properties.
In this paper we examine the ring of integer-valued polynomials on a subset of a Krull domain, and the divisor closed submonoid generated by a single polynomial . If doesn’t have any isolated points in any of the topologies given by essential valuations of , we can construct a divisor homomorphism from to a finite direct sum of copies of [Theorem 5.4]. This implies that is a Krull monoid, and hence, that is monadically Krull.
For the purpose of constructing divisor homomorphisms on monoids of integer-valued polynomials, we will study “relative” polynomial closure, that is, polynomial closure with respect to a subset of , in section 2. This modification of the concept of polynomial closure makes it possible to find finite polynomially dense subsets of arbitrary sets in section 3. Equipped with these finite polynomially dense sets we construct the actual divisor homomorphisms and, in some cases, divisor theories, to finite sums of copies of in sections 4 and 5.
A short review of integer-valued polynomial terminology: Let be a domain with quotient field , and . is called integer-valued if and is called integer-valued on if . If there are several possibilities for , we say -valued on instead of integer-valued on .
The ring of integer-valued polynomials on is written , and the ring of integer-valued polynomials on a subset of the quotient field of is denoted by :
Let be a domain with quotient field , and . The divisor-closed submonoid of generated by , which we write , is the multiplicative monoid consisting of all for which there exists and , such that .
Keep in mind that the elements of are not just polynomials in that divide some power of in . The co-factor is also required to be in . We will frequently use the following divisibility criterion for .
Let be the divisor closed submonoid of as in Definition 1.1 and . Then divides in if and only if divides in and the cofactor is in .
Multiplying a polynomial in by a constant in does not in general result in an element of . We can multiply elements of by some suitable constants, though.
Let be the valuation domain of a valuation on , , and the divisor-closed submonoid of generated by . Let and . If then .
Let and such that . Then both and are in , and . ∎
We recall the definitions of ideal content and fixed divisor, whose interplay will be an important ingredient of proofs. Let be a domain and . The content of , denoted , is the fractional ideal generated by the coefficients of . If is a principal ideal domain, we identify, by abuse of notation, ideals by their generators and say that is the gcd of the coefficients of . A polynomial is called primitive if , that is, in the case of a PID, if .
Let be a domain with quotient field , and . The fixed divisor of on , denoted , is the -submodule of generated by the image . Note that is a fractional ideal. If , we write for . If is a PID, we will, by abuse of notation, sometimes write a generator to stand for the ideal, e.g., for . A polynomial is called image-primitive if .
For polynomials in , image-primitive implies primitive, but not vice versa. One difference between ideal content and fixed divisor is that the ideal content is multiplicative for sufficiently nice rings (called Gaussian rings), including principal ideal rings, whereas the fixed divisor is not multiplicative. contains , but the containment can be strict.
Two easy but useful facts:
If is image-primitive then is image-primitive for all .
If is image-primitive then all divisors in of are also image-primitive.
In case is an intersection of valuation rings, then every is also in for all these valuation rings, and may be image-primitive as an element of , but not as an element of . In this case, we write
and write to express that is image-primitive when regarded as an element of .
Regarding valuation terminology: we use additive valuations, that is, a valuation is a map , where is a totally ordered group, satisfying
and we set . The valuation ring of a valuation on a field is and the valuation group is the image of in .
2. Relative polynomial closure
Definition 2.1 (relative polynomial closure).
Fix a domain with quotient field . Let and .
The polynomial closure of relative to is
If , and we call polynomially dense in relative to .
The definition of polynomial closure and polynomial density depends on the choice of . If there is any doubt about , we say -polynomial closure and -polynomially dense.
Polynomial closure relative to is the “usual” polynomial closure, introduced by Gilmer  and studied by McQuillan , the present author , Cahen , Park and Tartarone  and Chabert , among others. The reason why we generalize the well-known concept of polynomial closure will become apparent in the next section: when we consider polynomial closure relative to a set of polynomials whose irreducible factors are restricted to a finite set, it becomes possible to find finite polynomially dense subsets of any fractional set.
The following properties of polynomial closure relative to a subset of are easy to check.
Polynomial closure relative to is a closure operator, in the sense that
Polynomial closure relative to is the closure given by a Galois correspondence that maps every subset of to a subset of , and every subset of to a subset of , namely,
If then .
If is polynomially dense in relative to , and , then is polynomially dense in relative to .
When the domain is a valuation ring, then polynomially dense subsets of relative to are easily characterized:
Let be a valuation on a field , its valuation ring, and . Consider
is -polynomially dense in relative to .
(1) implies (2). If is closed under multiplication by non-zero constants in then (2) implies (1).
For every polynomial , . Therefore, by , and hence .
For every , , since . If and are such that , pick with . Then , but , so is not -polynomially dense in relative to . ∎
3. Finite polynomially dense subsets
Let be a finite set of irreducible polynomials in and the multiplicative submonoid of generated by and the non-zero constants of . That is, consists of all non-zero polynomials in whose irreducible factors in are (up to multiplication by non-zero constants) in .
We will now construct, for every subset of a discrete valuation ring , a finite polynomially dense subset of relative to . It is possible to admit fractional subsets of , but for simplicity’s sake we restrict ourselves to subsets of .
By discrete valuation, we mean, more precisely, a discrete rank valuation, that is, a valuation whose value group is isomorphic to . A normalized discrete valuation is one whose value group is actually equal to . The valuation ring of a discrete valuation is called discrete valuation ring, abbreviated DVR. As we all know, a DVR is a local principal ideal domain.
Let be a discrete valuation on with valuation ring , , and a finite-dimensional field extension over which splits. Let be an extension of to (), the valuation ring of and its maximal ideal. Say splits as with for and for over .
Then for all ,
This follows from the fact that whenever . ∎
Let be a topological space and . An isolated point of is an element having a neighborhood such that .
Let be a discrete valuation on and its valuation ring. Let be a finite set of monic irreducible polynomials in and the set of those polynomials in whose monic irreducible factors are all in . Let .
Then there exists a finite subset such that
and every such is, in particular, a finite set that is polynomially dense in relative to .
If no root of any is an isolated point of in -adic topology, then the above set can be chosen such as not to contain any root of any .
Let be the splitting field of over , an extension of to and the valuation ring of . Let be the set of distinct roots of polynomials of in . Then in (1) and (2) can be chosen with .
Let , , , and as in (3). Let be the maximal ideal of . We call the elements of “the roots”. We may assume and (otherwise the claimed facts are trivial). In view of Remark 3.1 it suffices to construct a set such that, for every finite sequence in ,
We will do this by constructing a finite covering of by disjoint sets and for each choosing a representative such that for every and every . This representative then satisfies , by Remark 3.1. If we take to be the set of representatives of covering sets then for every , is realized by some . By Lemma 2.3, this makes polynomially dense in relative to .
For any ideal of , we call a residue class “relevant” if .
We construct , () and inductively. Before step , initialize , , .
At the beginning of step , is a finite set of relevant residue classes of various with while is a finite set of relevant residue classes of each containing at least one root. In step , initialize ; then go through each and process it as follows:
If with then put in and in . Note that in this case is a -adic neighborhood of whose intersection with is , and that therefore is an isolated point of .
Else, if contains a relevant residue class of which doesn’t contain a root, pick such a , add a representative of to ; then put in .
Else place all relevant residue classes of contained in (each containing a root, by construction) in .
If is empty at the end of step , stop. Otherwise proceed to step .
Note that after each step , is a covering of . When the algorithm terminates with , then is a covering of and contains for each a representative satisfying for all . Therefore for all by Remark 3.1.
The algorithm terminates when no root is left in . For each root , one can give an upper bound on such that is no longer in . Namely, let such that for all roots . If then a residue class containing has been dropped as not relevant at or before step , so . If , then a residue class containing is placed in at step or earlier. Otherwise, contains an element of other than . Let , with minimal. Then will be placed in by step .
This shows (1). For (2), note that the set thus constructed contains no root of any except such as are isolated points of in -adic topology. For (3), note that every time an element is added to , a set containing at least one root is transferred from to and the number of roots in decreases. ∎
Thanks to the anonymous referee for pointing out that parts (1) and (2) of Proposition 3.3 can be show more quickly by applying Dickson’s theorem [Thm. 1.5.3], which says that the set of minimal elements of any subset of is finite and that for every there exists a minimal element with , to the subset of .
4. Divisor theories for monoids of integer-valued polynomials on discrete valuation rings
A short review of monoid terminology used in the definition of divisor homomorphism: By monoid we mean a semigroup that has a neutral element. All monoids we consider here are commutative, and they are cancellative, that is, whenever or , it follows that .
Let be a commutative monoid, written additively, and .
We say that divides in , and write , whenever there exists such that .
We call an element a greatest common divisor, abbreviated gcd, of a subset , if
for all : if for all then .
A monoid homomorphism is called a divisor homomorphism if in implies in .
A divisor homomorphism is called a divisor theory if each of the basis vectors (having in the -th coordinate and zeros elsewhere) occurs as gcd of a finite set of images .
We are preparing to construct divisor homomorphisms from certain submonoids of , where is a Krull domain, to finite sums of copies of , relating divisibility in to divisibility in a finitely generated free commutative monoid, which a priori looks much simpler. If is a direct sum of copies of , then the divisibility relation in is just the partial order given by the order relations on each component: Let with and . Then in is equivalent to for all . Therefore, any set of elements of has a unique gcd, namely, .
In what follows, we denote the normalized discrete valuation on corresponding to an irreducible polynomial by ; for , is the exponent to which occurs in the essentially unique factorization of in into irreducible polynomials, and for , .
In this section we examine the special case , where is a discrete valuation ring (DVR).
Let be a normalized discrete valuation on and its valuation ring. Let be a finite set of pairwise non-associated irreducible polynomials in and the multiplicative submonoid of generated by and the non-zero constants in . Let such that no root of any is an isolated point of in -adic topology. Let .
There exists a finite subset of that is polynomially dense in relative to and contains no root of any ; and for every such
is a divisor homomorphism. If is chosen minimal, is a divisor theory.
The existence of a finite polynomially dense subset containing no root of any is Proposition 3.3. Once we have a finite dense set, a minimal dense set can be obtained by removing redundant elements.
is clearly a monoid homomorphism. Now suppose such that , and set . We must show .
means for all and for all . The first shows , and therefore , and the second shows that for all . Since is polynomially dense in relative to , it follows that . We have shown to be a divisor homomorphism.
It remains to show that every for any and every for any occurs as the gcd of a finite set of images of elements of , provided is minimal.
We may assume, without changing , or in any way, that the elements of are in and primitive.
First, let be a generator of the maximal ideal of . The constant polynomial is an element of satisfying for all and for all .
Second, we note that every polynomial is an element of satisfying and for every .
Third, we show that for every , there exists such that and for all . We use the minimality of and Lemma 2.3: Since is polynomially dense in relative to , but is not, there exists a polynomial with and for all . Let be such a polynomial and . Then has the desired properties.
Fourth, we show that for every and there exists such that and . Let be any polynomial in with . If , set
Now for any and ,
5. Divisor homomorphisms on monadic monoids of integer-valued polynomials
What we have found out about the submonoid of consisting of polynomials whose irreducible factors in come from a fixed finite set, we now apply to the divisor closed submonoid of generated by a single polynomial.
Recall that , the divisor-closed submonoid of generated by , is the multiplicative monoid consisting of all those which divide some power of in . Also, it will be useful to recall the definition of image-primitive, and of , the fixed divisor of on from Definition 1.4.
First let us get a trivial case out of the way:
Let be a DVR, and with . Let be a set of primitive polynomials in representing the different irreducible factors of in . Let be the multiplicative submonoid of generated by and the units of . Then
Every element of is in , is primitive, and satisfies .
If , then divides in if and only if divides in .
is a divisor theory.
We will show (1) and (2). The remaining statements follow from (1).
is image-primitive on and hence primitive. The same holds for all powers of and for all divisors in of any power of by Remark 1.5. Therefore every divisor in of any power of is in , and vice versa, every element of is a divisor in of some power of . Therefore every element of is image-primitive on , and also .
Now let . Let and with . Then and with and . Since and are image-primitive on , we must have and . Since is primitive, . It follows that and therefore . ∎
Let be a domain with quotient field , a subset of , and . Let be a set of representatives (up to multiplication by a non-zero constant) of the irreducible factors of in . For instance, could be the set of monic irreducible factors of in . Or, in case that is a principal ideal domain, such as, for instance, a discrete valuation domain, can be chosen to be the set of primitive irreducible polynomials in dividing in . By we denote the multiplicative submonoid of generated by and the constants in . (Note that depends only on , not on the choice of ). Obviously . We now examine when the equality holds. In this non-trivial case we can give a divisor theory of [Theorem 5.3]. Otherwise, we have to be content with a divisor homomorphism [Theorem 5.4].
Let be a discrete valuation domain with quotient field , and . Let be multiplicative submonoid of generated by the irreducible factors of and the non-zero constants. If then
Let be the set of primitive irreducible polynomials in that divide in . Let with the content of and primitive. For arbitrary , we show that .
Let . Since , and we may apply the Archimedean axiom. Let such that .
Then , and both and are in . Therefore .
Now that for arbitrary , all factors of in are in . Therefore, all primitive irreducible factors of and all non-zero constants of , and furthermore, all products of such elements, are in . Finally, by Lemma 1.3, we can multiply elements of by any constant with , as long as the result is integer-valued on . Therefore, .
The reverse inclusion is trivial. ∎
Let be a normalized discrete valuation on and its valuation ring. Let and , such that no root of is an isolated point of in -adic topology. Let be the set of different monic irreducible factors of in and the multiplicative submonoid of generated by and the non-zero constants in . By denote the divisor-closed submonoid of generated by .
There exists a finite polynomially dense subset of relative to that does not contain any root of ; and for every such
is a divisor homomorphism. If and is chosen minimal then is a divisor theory.
Recall that a Krull domain is a domain satisfying the following conditions with respect to , the set of prime ideals of height :
For every , the localization is a DVR.
Each non-zero lies in only finitely many .
If is a Krull domain, we denote the normalized discrete valuation on the quotient field of whose valuation ring is by . Such a valuation is called an essential valuation of the Krull domain .
Again, the existence of finite -polynomially dense subsets of relative to in the following theorem is guaranteed by Proposition 3.3.
Let be a Krull domain with quotient field and such that doesn’t have any isolated points in -adic topology for any essential valuation of . Let , and the divisor-closed multiplicative submonoid of generated by .
Let be the finite set of different monic irreducible factors of in and the multiplicative submonoid of generated by and the non-zero constants.
Let be the finite set of primes of height of such that either or and . For each , let be a finite subset of that is -polynomially dense relative to in and contains no root of .
is a divisor homomorphism.
It is clear that is a monoid homomorphism. Now assume with . It suffices to show that divides in and that the co-factor is in for all , because then , which implies that divides in by Remark 1.2.
Let . That is in follows from for all irreducible factors of and in .
Consider a prime of height of that is not in . For such a prime, and is image-primitive in . We may apply Lemma 5.1 (3) and deduce that .
Now for , let be the projection of onto , and call the latter monoid . From it follows that divides . Let be the divisor closed submonoid of generated by . Then is a submonoid of , and is the restriction to of the divisor homomorphism in 4.2. Now the fact that divides implies , by Proposition 4.2. ∎
Let be a Krull domain and a subset that doesn’t have any isolated points in any of the topologies given by essential valuations of . Let . Then , the divisor closed submonoid of generated by , is a Krull monoid.
In particular, for every Krull domain and every , the divisor closed submonoid of generated by is a Krull monoid.
Monoids with the property that the divisor closed submonoid generated by any single element is a Krull monoid have been called monadically Krull by A. Reinhart. Without using divisor homomorphisms, through an approach completely different from ours, Reinhart showed that is monadically Krull whenever is a principal ideal domain [Thm. 5.2].
Corollary 5.5 generalizes Reinhart’s result to Krull domains, and to integer-valued polynomials on sufficiently nice subsets. The explicit divisor homomorphisms of Proposition 4.2 and Theorems 5.3 and 5.4 give additional information on the arithmetic of submonoids of . It remains an open problem to find the precise divisor theories (cf. Def. 4.1) of those monoids of integer-valued polynomials for which the above theorems provide divisor homomorphisms.
Acknowledgment. Enthusiastic thanks go out to the anonymous referee for his/her meticulous reading of the paper and the resulting corrections.
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