Polynomial functions on subsets of non-commutative rings — a link between ringsets and null-ideal sets

Regarding polynomial functions on a subset S of a non-commutative ring R, that is, functions induced by polynomials in R[x] (whose variable commutes with the coefficients), we show connections between, on one hand, sets S such that the integer-valued polynomials on S form a ring, and, on the other hand, sets S such that the set of polynomials in R[x] that are zero on S is an ideal of R[x]. 2010 Mathematics Subject Classification: 13F20, 16D25, 16P10, 16S99


Introduction
In the theory of polynomial mappings on commutative rings, there are two notable subtopics, namely, polynomial functions on finite rings, and rings of integer-valued polynomials.Here, we are concerned with generalizations of these two topics to polynomial mappings on noncommutative rings, as proposed by Loper and Werner [8], and developed further by Werner [11][12][13][14], Peruginelli [9,10], and the present author [3][4][5], among others.More particularly, we will investigate connections between null ideals of polynomials on finite non-commutative rings and integer-valued polynomials on non-commutative rings.
When we talk about polynomial functions on a non-commutative ring R, we mean functions induced by elements of the usual polynomial ring R[x] whose indeterminate x commutes with the elements of R. Non-commutative rings R for which polynomial functions have been studied include rings of quaternions [7,11,14], and matrix algebras [4,5,12].
To begin, we introduce the two objects we want to relate, null ideals and rings of integervalued polynomials, in their original, commutative setting: When considering polynomial functions on a finite commutative ring R, the first thing one likes to know is the so called null ideal N(R) of R[x] consisting of all null-polynomials, that is, polynomials such that the function induced on R by substitution of the variable is zero.Regarding integer-valued polynomials, they are defined, for a domain D with quotient field K, as those polynomials f in K[x] such that the polynomial function defined by f on K takes every element of D to an element of D [1].
We now generalize polynomial functions to non-commutative rings.
Let R be a (possibly non-commutative) ring and Then, f induces two polynomial functions on R, namely, the right polynomial function f r : R → R and the left polynomial function f l : R → R, where There are other generalizations of polynomial functions to polynomials with coefficients in non-commutative rings, using polynomials whose indeterminate does not commute with the coefficients.We are not concerned with this kind of generalized polynomials here.Our topic are left and right polynomial functions defined, as above, on a non-commutative ring by polynomials in the usual polynomial ring R[x] whose indeterminate x commutes with the elements of R.
Regarding these left and right polynomial functions on a non-commutative ring R, we notice that they do not, in general, admit a substitution homomorphism.
It may happen, for some s in R, and f, g ∈ R[x] that f r (s)g r (s) ( f g) r (s) and also f l (s)g l (s) ( f g) l (s).
In order to generalize null-ideals to polynomials on finite non-commutative rings, we consider the sets of right and left null-polynomials, respectively, on R. We note that, in the absence of a substitution homomorphism, neither set is necessarily an ideal of R[x].Definition 1.1.Let R a ring and f ∈ R[x].The polynomial f is called a right null-polynomial on R in case f r (s) = 0 for all s ∈ R, and a left null-polynomial on R in case f l (s) = 0 for all s ∈ R.
We denote the sets of right and left null polynomials on R, respectively, by It is immediately clear that Fact 1.2 (Werner [13]).For every ring R, The last expression is zero whenever Similarly, by interchanging left and right, we see that ITM Web of Conferences 20, 01003 (2018) https://doi.org/10.1051/itmconf/20182001003ICM 2018 Whether N r (R) is also a right ideal of R[x], and thus an ideal, for any finite ring R, is an open question, and similarly the question whether N l (R) is a left ideal and therefore an ideal.There are no known counterexamples.
Werner has found many sufficient conditions on R for N r (R) to be a right ideal [13], but none of them are necessary.If we take R as the ring of upper triangular matrices over a commutative ring T , we can, by judicious choice of T , find examples of rings violating all of Werner's necessary conditions, which nevertheless satisfy that N r (R) is a right ideal and N l (R) is a left ideal of R[x] [5].Such examples can also be found among rings of integer-valued polynomials on quaternions [14].Now, when we generalize integer-valued polynomials to polynomials with coefficients in a non-commutative ring, the usual setup (as introduced in [4]) is Definition 1.3.Let D be a domain with quotient field K and A a finitely generated, torsionfree D-algebra.Let B = A ⊗ D K. To avoid certain pathologies, we stipulate that A ∩ K = D when A and K are canonically embedded in B.
Then, the set of right integer-valued polynomials (with coefficients in B) on and the set of left integer-valued polynomials (with coefficients in B) on A is We remark that it is not a priori clear that Int r B (A) and Int l B (A) are rings, because, in the absence of a substitution homomorphism, closure under multiplication is not a given.M n (K) (M n (D)) coincides with Int r M n (K) (M n (D)) (shown to be a ring by Werner [12]), and is canonically isomorphic to M n (Int K (M n (D))) [4].The algebras for which Int B (A) Int K (A) ⊗ D A thus holds have been characterized by Peruginelli and Werner [10].
For T n (D) the ring of n × n upper triangular matrices with entries in D, Int r T n (K) (T n (D)) is isomorphic to the algebra of matrices whose entries in position ( j, k) are in Int K (T n−k+1 (D)), and Int l T n (K) (T n (D)) is isomorphic to the algebra of matrices whose entries in position ( j, k) are from Int K (T j (D)) [5].The commutative rings of integer-valued polynomials on upper (or lower) triangular matrices (with coefficients in K), Int K (T n (D)), are of interest in their own right [2].
Again, Werner has given different sufficient conditions on A for Int r B (A) to be a ring, but we know that these conditions are not necessary.Taking A as the ring of upper triangular matrices over a judiciously chosen domain D we can find examples where Int r B (A) and Int l B (A) are rings, but all known sufficient conditions are violated [5].Also, such examples can be found among rings of integer-valued polynomials over quaternion algebras [14].

A connection between ringsets and null-ideal sets
We do not know whether Int r B (A) is always closed under multiplication; nor do we know whether N r (R) is always an ideal of R[x].As a way out of this quandary, we widen the scope of our investigation.Following Werner [14], we consider integer-valued polynomials on subsets of A.
ITM Web of Conferences 20, 01003 (2018) https://doi.org/10.1051/itmconf/20182001003ICM 2018 Here, in addition to integer-valued polynomials on subsets, we will also consider nullpolynomials on subsets, and demonstrate a connection between the two.
In what follows, we will often confine ourselves to right polynomial functions, with the understanding that everything also holds, mutatis mutandis, for left polynomial functions.In the context of right polynomial functions, f (c) means f r (c).Likewise, the set of left integer-valued polynomials on S is , A) is closed under multiplication, and hence a ring.
It is easy to give examples, both of ringsets and of sets that are not ringsets: For any D-algebra satisfying that d∈D\{0} dA = (0), which is, for instance, the case if A is a free Dmodule and D is a Noetherian or Krull domain, Werner [14] showed that a singleton {s} ⊆ A is a ringset if and only if s is in the center of A. To see "only if," suppose that s does not commute with some t ∈ A. Let d ∈ D \ {0} such that ts − st dA, and let Note that an arbitrary union of ringsets is always a ringset, by the fact that an intersection of rings is a ring.Definition 2.2.Let R be a ring and S a subset of R. We denote by N r R (S ) the set of polynomials f ∈ R[x] such that for all s ∈ S , f r (s) = 0. We abbreviate N r R (R) by N r (R).Likewise, we denote by N l R (S ) the set of polynomials f ∈ R[x] such that for all s ∈ S , f l (s) = 0 and abbreviate . This is demonstrated, just like the fact that N r (R) is a left ideal, by equation 1.
The question is: for which sets S is N r R (S ) a right ideal?Likewise, N l R (S ) is always a right ideal of R[x], and the question is: for which sets S is N l R (S ) a left ideal?Definition 2.4.We say that S (as a subset of R) is a right null-ideal set if N r R (S ) is a right ideal of R[x], and hence an ideal of R [x].
We say that S (as a subset of R) is a left null-ideal set if and hence an ideal of R[x].
We will now give a criterion for ringsets in terms of null-ideal sets.For this purpose, we introduce null-polynomials modulo an ideal.We will later rephrase everything using null polynomials in the strict sense.We illustrate the principle of treating null-ideals and rings of integer-valued polynomials in one common setting by another one of Werner's sufficient conditions.Therefore, pol r R (S , I) is closed under multiplication from the right by units of T .Also, pol r R (S , I) is certainly closed under multiplication from the right by elements in the center of T (thanks to the fact that S is a subset of T ), and closed under addition and subtraction.Since every element of T is a finite sum of products of central elements and units of T , we may conclude that pol r R (T, I) is closed under multiplication from the right by elements of T .Corollary 3.6 (Werner [14,Prop. 6.13]).If A is generated by units as an algebra over its center and S ⊆ A is closed under conjugation by units of A, then Int r B (S , A) is a ring, i.e., S is a right ringset.Corollary 3.7.If R is generated by units as an algebra over its center and S ⊆ R is closed under conjugation by units of R, then N r R (S ) is an ideal of R[x], i.e., S is a right null-ideal set.
The null ideal is important, because the residue classes of R[x] mod N(R) correspond to the different polynomial mappings on R and hence the index [R[x] : N(R)] indicates the number of different polynomial mappings on R. ITM Web of Conferences 20, 01003 (2018) https://doi.org/10.1051/itmconf/20182001003ICM 2018 Yet, there are no known counterexamples.Whether Int r B (A) and Int l B (A) are rings for any D-algebra A as in Definition 1.3 remains an open question.In some cases it is possible to describe Int r B (A) and Int l B (A) via their relation to the commutative ring Int K (A) = Int r B (A)∩ K[x] = Int l B (A)∩ K[x], for instance, when A = M n (D) is the ring of n × n matrices over D. Here, Int l

Definition 2 . 1 .
Let A be a D-algebra and everything as in Definition 1.3 and S ⊆ A. The set of right integer-valued polynomials on S is Int r B (S , A) = { f ∈ B[x] | ∀s ∈ S f r (s) ∈ A}. S is called a right ringset if Int B (S , A) is closed under multiplication, and hence a ring.

Definition 2 . 5 .
Let R be a ring and S a subset of R and I an ideal of R.A polynomial f ∈ R[x] is called a right null polynomial modulo I on S if f r (s) ∈ I for every s ∈ S .A polynomial f ∈ R[x] is called a left null-polynomial modulo I on S if f l (s) ∈ I for every s ∈ S .We denote by N r (R mod I) (S ) the set of right null-polynomials mod I on S and byN l (R mod I) (S )the set of left null-polynomials mod I on S .ITM Web of Conferences 20, 01003 (2018) https://doi.org/10.1051/itmconf/20182001003ICM 2018

Theorem 3 . 5 .
Let R be a ring, T a subring of R and I an ideal of T .If T is generated by units as an algebra over its center, then 1. pol r R (T, I) is a right T -module.2.More generally, for every subset S of T that is closed under conjugation by units of T ,pol r R (S , I) is a right T -module.Proof.Let f = k c k x k ∈ pol r R (T, I), and u a unit of T .Then f u ∈ pol r R (S , I), because, for any t ∈ S , t can be written as t = u −1 τu with τ = utu −1 ∈ S , and then( f u)(t) = k c k ut k = k c k uu −1 τ k u = f (τ)u,where f (τ)u ∈ I, because f (τ) ∈ I and I is an ideal of T .
R (T, I) a right T -module (with the restricion of the multiplication of R[x] as scalar multiplication)?