Unique factorization in polynomial rings with zero divisors

Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

On Zero Left Prime Factorizations for Matrices over Unique Factorization Domains

In this paper, zero prime factorizations for matrices over a unique factorization domain are studied. We prove that zero prime factorizations for a class of matrices exist. Also, we give an algorithm to directly compute zero left prime factorizations for this class of matrices.

A Signature-Based Algorithm for Computing Gröbner Bases over Principal Ideal Domains

Signature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller’s algorithm (J Symb Comput 6(2–3), 345–359, 1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).

Simple derivations on tensor product of polynomial algebras

Let [Formula: see text] be an unique factorization domain containing a field [Formula: see text] of characteristic zero and let [Formula: see text] and [Formula: see text] be two [Formula: see text]-algebras. Let [Formula: see text] and [Formula: see text] be two generalized triangular [Formula: see text]-derivations of [Formula: see text] and [Formula: see text], respectively. Denote the unique [Formula: see text]-derivation [Formula: see text] of [Formula: see text] by [Formula: see text]. Then with some conditions on [Formula: see text] and [Formula: see text], it is shown that [Formula: see text] is a simple derivation of [Formula: see text] if and only if [Formula: see text] is [Formula: see text]-simple and [Formula: see text] is [Formula: see text]-simple. We also show that if [Formula: see text] and [Formula: see text] are positively homogeneous derivations and [Formula: see text] is a generalized triangular derivation, then [Formula: see text] is simple derivation of [Formula: see text] if and only if [Formula: see text] is a simple derivation of [Formula: see text] and [Formula: see text] is a simple derivation of [Formula: see text].

Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (II)

<p>Let 1 &lt; s₁ &lt; . . . &lt; s_k be integers, and assume that κ ≥ 2 (so s_k ≤ 3). Then there exists a local UFD (Unique Factorization Domain) (R,M) such that:</p><p>(1) Height(M) = s_k.</p><p>(2) R = R' = ∩{VI (V,N) € V_j}, where V_j (j = 1, . . . , κ) is the set of all of the Rees valuation rings (V,N) of the M-primary ideals such that trd((V I N) I (R I M)) = s_j - 1.</p><p>(3) With V₁, . . . , V_κ as in (2), V₁ ∪ . . . V_κis a disjoint union of all of the Rees valuation rings of allof the M-primary ideals, and each M-primary ideal has at least one Rees valuation ring in each V_j .</p>

A characterization of p-bases of rings of constants

We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.

PRESENTATIONS AND MODULE BASES OF INTEGER-VALUED POLYNOMIAL RINGS

Let D be an integral domain with quotient field K. For any set X, the ring Int (DX) of integer-valued polynomials onDX is the set of all polynomials f ∈ K[X] such that f(DX) ⊆ D. Using the t-closure operation on fractional ideals, we find for any set X a D-algebra presentation of Int (DX) by generators and relations for a large class of domains D, including any unique factorization domain D, and more generally any Krull domain D such that Int (D) has a regular basis, that is, a D-module basis consisting of exactly one polynomial of each degree. As a corollary we find for all such domains D an intrinsic characterization of the D-algebras that are isomorphic to a quotient of Int (DX) for some set X. We also generalize the well-known result that a Krull domain D has a regular basis if and only if the Pólya–Ostrowski group of D (that is, the subgroup of the class group of D generated by the images of the factorial ideals of D) is trivial, if and only if the product of the height one prime ideals of finite norm q is principal for every q.

ON BOUNDING MEASURES OF PRIMENESS IN INTEGRAL DOMAINS

Let D be an integral domain. In this paper, we investigate two (integer- or ∞-valued) invariants ω(D, x) and ω(D) which measure how far a nonzero x ∈ D is from being prime and how far an atomic integral domain D is from being a unique factorization domain (UFD), respectively. In particular, we are interested in when there is a nonzero (irreducible) x ∈ D with ω(D, x) = ∞ and the relationship between ω(A, x) and ω(B, x), and ω(A) and ω(B), for an extension A ⊆ B of integral domains and a nonzero x ∈ A.