Monoids characterized by their quasi-injective S-systems
type: Item
instance of: scholarly article
title: Monoids characterized by their quasi-injective S-systems
zbMATH Open document ID: 0625.20057
DOI: 10.1007/BF02575022
published in: Semigroup Forum
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134902
review text: Let S be a monoid. A right S-system \(M_ S\) is called finitely injective if for every S-monomorphism \(f: X_ S\to Y_ S\), where X is a finitely generated S-system, and for every S-homomorphism \(g: X_ S\to M_ S\), there exists an S-homomorphism \(h: Y_ S\to M_ S\) such that \(hf=g\). \(M_ S\) is quasi-injective if for \(N_ S\subseteq M_ S\) and every S- homomorphism \(f: N_ S\to M_ S\), there exists an S-homomorphism \(g: M_ S\to M_ S\) such that \(g|_ N=f\). M is countably \(\Sigma\)- injective if the direct sum of countably many copies of M is injective. S is right noetherian if and only if S satisfies the ascending chain condition for right ideals. For a monoid S, the following conditions are shown to be equivalent in the category of centered S-systems: (1) Each direct sum of injective S-systems is injective. (2) Each direct sum of weakly injective S-systems is weakly injective. (3) Each injective S- system is countably \(\Sigma\)-injective. (4) Each finitely injective S- system is weakly injective. (5) S is noetherian. Furthermore, it has been proved that each direct sum of quasi-injective S-systems is quasi- injective if and only if S is noetherian and each quasi-injective S- system is injective.
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 4015022
zbMATH Keywords: right S-system
zbMATH Keywords: finitely generated S-system
zbMATH Keywords: right ideals
zbMATH Keywords: category of centered S-systems
zbMATH Keywords: direct sum of injective S-systems
zbMATH Keywords: weakly injective S-systems
zbMATH Keywords: quasi-injective S-systems
Wikidata QID: Q114233781
author: Javed Ahsan
MaRDI profile type: Publication
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cites work: Q4161512
cites work: Quasi-injective $S$-systems and their $S$-endomorphism semigroup
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cites work: Q3866309
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description: scientific article; zbMATH DE number 4015022