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We give examples of atomic integral domains satisfying each of the eight logically possible combinations of existence or non-existence of the following kinds of elements: 1) primes, 2) absolutely irreducible elements that are not prime, and 3) irreducible elements that are not absolutely irreducible. A non-zero non-unit is called absolutely irreducible (or, a strong atom) if every one of its powers factors uniquely into irreducibles., Comment: 21 pages

Let $V$ be a valuation ring of a global field $K$. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $V$, that is, an element of $\text{Int}(V) = \{ f \in K[X] \mid f(V) \subseteq V \}$, which has precisely $k$ essentially different factorizations into irreducible elements of $\text{Int}(V)$ whose lengths are exactly $n_1,\ldots,n_k$. In fact, we show more, namely that the same result holds true for every discrete valuation domain $V$ with finite residue field such that the quotient field of $V$ admits a valuation ring independent of $V$ whose maximal ideal is principal or whose residue field is finite. If the quotient field of $V$ is a purely transcendental extension of an arbitrary field, this property is satisfied. This solves an open problem proposed by Cahen, Fontana, Frisch and Glaz in these cases.

Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain $(R,M)$ with finite residue field, it is known that there exist absolutely irreducible elements, that is, irreducible elements all of whose powers factor uniquely, and non-absolutely irreducible elements. We completely and constructively characterize the absolutely irreducible elements among split integer-valued polynomials. They correspond bijectively to finite sets, which we call \emph{balanced}, characterized by a combinatorial property regarding the distribution of their elements among residue classes of powers of $M$. For each such balanced set as the set of roots of a split polynomial, there exists a unique vector of multiplicities and a unique constant so that the corresponding product of monic linear factors times the constant is an absolutely irreducible integer-valued polynomial. This also yields sufficient criteria for integer-valued polynomials over Dedekind domains to be absolutely irreducible., Comment: This version comes with the shortened introduction as it appears in the journal version. [v2] contains an extended introduction, the remaining sections are identical to [v3]

We describe the prime ideals and, in particular, the maximal ideals in products $R = \prod D_\lambda$ of families $(D_\lambda)_{\lambda \in \Lambda}$ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra $\prod \mathcal{P}(\max(D_\lambda))$, where $\max(D_\lambda)$ is the spectrum of maximal ideals of $D_\lambda$, and $\mathcal{P}$ denotes the power set. If every $D_\lambda$ is in a certain class of rings including finite character domains and one-dimensional domains, we completely characterize the maximal ideals of $R$. If every $D_\lambda$ is a Pr\"ufer domain, we completely characterize all prime ideals of $R$., Comment: to appear in Communications in Contemporary Mathematics

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring $\text{Int}(D)=\{f\in K[x]\mid f(D)\subseteq D\}$, of integer-valued polynomials on a principal ideal domain $D$ with quotient field $K$, we give an easy to verify graph-theoretic sufficient condition for an element to be absolutely irreducible and show a partial converse: the condition is necessary and sufficient for polynomials with square-free denominator., Comment: To appear in Comm. Algebra

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 $R[x]$., Comment: 9 pages, conference paper for "advances in algebra ..." at Ton Duc Thang University, Vietnam, Dec 18-20, 2018

Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of finite index, and $K$ its quotient field. Let $\operatorname{Int}(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ be the ring of integer-valued polynomials on $D$. Given any finite multiset $\{k_1, \ldots, k_n\}$ of integers greater than $1$, we construct a polynomial in $\operatorname{Int}(D)$ which has exactly $n$ essentially different factorizations into irreducibles in $\operatorname{Int}(D)$, the lengths of these factorizations being $k_1$, \ldots, $k_n$. We also show that there is no transfer homomorphism from the multiplicative monoid of $\operatorname{Int}(D)$ to a block monoid.

There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coefficients on an algebra of upper triangular matrices is an ideal., Comment: to appear in Monatsh. Math; 15 pages

Let $D$ be a domain and $M$ a maximal ideal of $D$. The ring of integer-valued polynomials on a subset $E$ of $D$, as well as more general rings of functions from $E$ to $D$, can be viewed as subrings of the product $D^E=\prod_{e\in E}D$. We investigate which maximal ideals of $Int(E,D)$ (or any other subring of $D^E$) come from ultrapowers of the residue class ring $D/M$., Comment: 12 pages, to appear in J. Pure Appl. Algebra

Let D be a Krull domain and Int(D) the ring of integer-valued polynomials on D. For any f in Int(D), we explicitly construct a divisor homomorphism from [f], the divisor-closed submonoid of Int(D) generated by f, to a finite sum of copies of (N_0,+). This implies that [f] is a Krull monoid. For V a discrete valuation domain, we give explicit divisor theories of various submonoids of Int(V). In the process, we modify the concept of polynomial closure in such a way that every subset of D has a finite polynomially dense subset. The results generalize to Int(S,V), the ring of integer-valued polynomials on a subset, provided S doesn't have isolated points in v-adic topology., Comment: 12 pages; v.2 contains corrections, in that some necessary conditions on those subsets S, for which we consider integer-valued polynomials on subsets, are imposed

Let D be a domain with quotient field K and A a D-algebra. We call a polynomial with coefficients in K that maps every element of A to an element of A "integer-valued on A". For commutative A we also consider integer-valued polynomials in several variables. For an arbitrary domain D and I an arbitrary ideal of D we show I-adic continuity of integer-valued polynomials on A. For Noetherian one-dimensional D, we determine the spectrum and Krull dimension of the ring Int_D(A) of integer-valued polynomials on A. We do the same for the ring of polynomials with coefficients in M_n(K), the K-algebra of n x n matrices, that map every matrix in M_n(D) to a matrix in M_n(D)., Comment: 17 pages; a glitch in the published version (J.Algebra 373 (2013) 414-425) has been corrected in this post-preprint, namely, in Prop. 6.2 and Thm. 6.3, the assumption "zero Jacobson radical" needs to be replaced by the stronger assumption "intersection of maximal ideals of finite index is zero"

For an arbitrary finite set S of natural numbers greater 1, we construct an integer-valued polynomial f, whose set of lengths in Int(Z) is S. The set of lengths of f is the set of all natural numbers n, such that f has a factorization as a product of n irreducibles in Int(Z)={g in Q[x] | g(Z) contained in Z}., Comment: To appear in Monatshefte f\"ur Mathematik; 11 pages

We describe the Sylow $p$-groups of the group of polynomial permutations of the integers mod $p^n$.

A Pythagorean n-tuple is an integer solution of x_1^2+...+x_{n-1}^2=x_n^2. For n=4 and n=6, the Pythagorean n-tuples admit a parametrization by a single n-tuple of polynomials with integer coefficients (which is impossible for n=3). For n=5, there is an integer-valued polynomial Pythagorean 5-tuple which parametrizes Pythagorean quintuples (similar to the case n=3). Pythagorean quadruples are closely related to (integer) Descartes quadruples (solutions of 2(b_1^2+b_2^2+b_3^2+b_4^2) = (b_1+b_2+b_3+b_4)^2), which we also parametrize by a Descartes quadruple of polynomials with integer coefficients., Comment: 11 pages; to appear in J. Pure Appl. Algebra; June 28 2011: corrected typo in Thm 1 (plus instead of minus in first coord)

We investigate non-unique factorization of polynomials in Z_{p^n}[x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, Z_{p^n}[x] is atomic. We reduce the question of factoring arbitrary non-zero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of Z_{p^n}[x] is a direct sum of monoids corresponding to irreducible polynomials in Z_p[x], and we show that each of these monoids has infinite elasticity. Moreover, for every positive integer m, there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles., Comment: 11 pages

Let f in Z[X,Y,Z] be a non-constant, absolutely irreducible, homogeneous polynomial with integer coefficients, such that the projective curve given by f=0 has a function field isomorphic to the rational function field Q(t). We show that all integral solutions of the Diophantine equation f=0 (up to those corresponding to some singular points) can be parametrized by a single triple of integer-valued polynomials. In general, it is not possible to parametrize this set of solutions by a single triple of polynomials with integer coefficients., Comment: Dedicated to Prof. W. Narkiewicz on the occasion of his 70th birthday. To appear in Functiones et Approximatio. 4 pages

If, for a subset S of Z^k, we compare the conditions of being parametrizable (a) by a single k-tuple of polynomials with integer coefficients, (b) by a single k-tuple of integer-valued polynomials and, (c) by finitely many k-tuples of polynomials with integer coefficients (variables ranging through the integers in each case) then (a) implies (b) (obviously), (b) implies (c), and neither converse holds. Condition (b) is equivalent to the set S being the set of integer values taken by some k-tuple of polynomials with rational coefficients as the variables range through the integers. We also show that every co-finite subset of Z^k is parametrizable a single k-tuple of polynomials with integer coefficients., Comment: to appear in Comm. Algebra

It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials., Comment: to appear in J. Pure Appl. Algebra