unique factorization domains
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2020 ◽  
Vol 2020 ◽  
pp. 1-3
Author(s):  
Jinwang Liu ◽  
Tao Wu ◽  
Dongmei Li ◽  
Jiancheng Guan

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.


Author(s):  
Gyu Whan Chang

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.


2020 ◽  
Vol 13 (1) ◽  
pp. 165-180
Author(s):  
Sarah M. Fleming ◽  
Susan Loepp

2019 ◽  
Vol 19 (08) ◽  
pp. 2050150
Author(s):  
Leila Benferhat ◽  
Safia Manar Elislam Benoumhani ◽  
Rachid Boumahdi ◽  
Jesse Larone

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.


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