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.

Unique factorization property of non-unique factorization domains

Let [Formula: see text] be an integral domain, [Formula: see text] be the polynomial ring over [Formula: see text], [Formula: see text] be the so-called [Formula: see text]-operation on [Formula: see text], and [Formula: see text]-Spec[Formula: see text] be the set of prime [Formula: see text]-ideals of [Formula: see text]. A nonzero nonunit of [Formula: see text] is said to be homogeneous if it is contained in a unique maximal [Formula: see text]-ideal of [Formula: see text]. We say that [Formula: see text] is a homogeneous factorization domain (HoFD) if each nonzero nonunit of [Formula: see text] can be written as a finite product of pairwise [Formula: see text]-comaximal homogeneous elements. In this paper, among other things, we show that (1) a Prüfer [Formula: see text]-multiplication domain (P[Formula: see text]MD) [Formula: see text] is an HoFD if and only if [Formula: see text] is an HoFD (2) if [Formula: see text] is integrally closed, then [Formula: see text] is a P[Formula: see text]MD if and only if [Formula: see text]-Spec[Formula: see text] is treed, and (3) [Formula: see text] is a weakly Matlis GCD-domain if and only if [Formula: see text] is an HoFD with [Formula: see text]-Spec[Formula: see text] treed. We also study the HoFD property of [Formula: see text] constructions, pullbacks, and semigroup rings.

Additive decompositions of polynomials over unique factorization domains

Additive decompositions over finite fields were extensively studied by Brawely and Carlitz. In this paper, we study the additive decomposition of polynomials over unique factorization domains.