Monoids over which all flat left acts are regular
type: Item
instance of: scholarly article
title: Monoids over which all flat left acts are regular
zbMATH Open document ID: 0855.20053
DOI: 10.1016/0022-4049(96)00023-0
published in: Journal of Pure and Applied Algebra
publication date: 1996-11-13T00:00:00Z
review text: Let \(S\) be a monoid. A left \(S\)-act \({_SA}\) is flat (strongly flat) if the functor \(A\otimes -\) preserves monomorphisms (pullbacks, respectively). A left \(S\)-act \(A\) is called regular if for any \(a\in A\) there exists a homomorphism \(f:Sa\to S\) such that \(f(a)a=a\). The main theorem states that the following conditions on a monoid \(S\) are equivalent: 1) All flat left \(S\)-acts are regular; 2) All weakly flat left \(S\)-acts are regular; 3) Every cyclic subact of every flat left \(S\)-act is strongly flat; 4) Every cyclic subact of every weakly flat left \(S\)-act is strongly flat; 5) Every element of \(S\) different from 1 is a left zero.
reviewed by: Peeter Normak
Mathematics Subject Classification ID: 20M50
zbMATH DE Number: 920029
zbMATH Keywords: regular acts
zbMATH Keywords: monoids
zbMATH Keywords: flat left acts
zbMATH Keywords: weakly flat left acts
zbMATH Keywords: cyclic subacts
author: Zhongkui Liu
MaRDI profile type: Publication
full work available at URL: 0022-4049(96)00023-0
OpenAlex ID: W2099154408
cites work: Pullback-Flat Acts are Strongly Flat
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description: scientific article; zbMATH DE number 920029