Some results on skew generalized power series rings
type: Item
instance of: scholarly article
title: Some results on skew generalized power series rings
zbMATH Open document ID: 1358.13023
author: Kamal Paykan
author: A. Moussavi
published in: Taiwanese Journal of Mathematics
publication date: 2017-03-09T00:00:00Z
review text: A partially ordered set \((S,\leq)\) is said to be Artinian if every strictly decreasing sequence of elements of \(S\) is finite and is said to be narrow if every subset of pairwise order-incomparable elements of \(S\) is finite. Let \(R\) be a ring and \((S,\leq)\) a strictly ordered monoid, both of them are not necessarily commutative. Let \(\omega: S\longrightarrow\mathrm{End }E\) a monoid homomorphism. For \(s\in S\), put \(\omega_s=\omega(s)\). Denote by \(RS,\omega\) the set of all the functions \(f:S\longrightarrow R\) such that the support \(\mathrm{supp}(f)=\{s\in S;f(s)\not=0\}\) is Artinian and narrow. Then for any \(s\in S\) and \(f,g\in RS,\omega\), the set \(X_s(f,g)=\{(x,y)\in \mathrm{supp}(f)\times\mathrm{supp}(g); s=xy\}\) is finite. Thus one can define the product \(fg:S\longrightarrow R\) as follows \(\displaystyle fg(s)=\sum_{(u,v)\in X_s(f,g)}f(u)\omega_ug(v)\). With pointwise addition and this multiplication, \(RS,\omega\) becomes a ring, called the ring of skew generalized power series with coefficients in \(R\) and exponents in \(S\). This kind of construction includes many classical ring constructions. In this paper, the authors study when \(RS,\omega\) has a (flat) projective socle and when it is local, semilocal, semiperfect, semiregular, left quasi-duo, clean, exchange, right stable range one, projective-free and I-ring.
reviewed by: Ali Benhissi
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16S99
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 6693688
zbMATH Keywords: skew generalized power series ring
zbMATH Keywords: (flat)projective socle
zbMATH Keywords: local
zbMATH Keywords: semilocal
zbMATH Keywords: semiperfect
zbMATH Keywords: semiregular
zbMATH Keywords: I-ring
zbMATH Keywords: quasi-duo ring
zbMATH Keywords: projective-free ring
MaRDI profile type: Publication
full work available at URL: tjm.21.2017.7327
OpenAlex ID: W2581679787
DOI: 10.11650/TJM.21.2017.7327
description: scientific article; zbMATH DE number 6693688