Weakly \(N\)-injective modules
type: Item
instance of: scholarly article
title: Weakly \(N\)-injective modules
zbMATH Open document ID: 0869.16002
author: O. Diekmann
published in: Journal of the Egyptian Mathematical Society
publication date: 1997-09-04T00:00:00Z
review text: Properties of weak \(N\)-injectivity as a generalisation of \(N\)-injectivity [see \textit{T. Albu} and \textit{C. Nastasescu}, Relative finiteness in module theory (Pure Appl. Math. 84, 1984; Zbl 0556.16001)] are studied. Given two right \(R\)-modules \(M\) and \(N\), we say that \(M\) is weakly \(N\)-injective if for every \(f:N\to E(M)\), there exists a monomorphism \(h:M\to E(M)\) and a homomorphism \(g:N\to M\) such that \(f=hg\). If \(R\) is a PID, let \(R_{p^\infty}\) denote the set of all \((\frac{a}{p^n})+R\), where \(a\in R\), \(n\in\mathbb{Z}\) and \(p\) is a prime in \(R\). A right \(R\)-module \(X\) is found to be weakly \(R_{p^\infty}\)-injective if and only if \(X\) is isomorphic to \((\oplus R_{p^\infty})\oplus Y\), where \(Y\) has no \(p\)-torsion element. Examples show that weak \(N\)-injectivity is indeed a generalization of \(N\)-injectivity. The class of all \(N\)-injective modules, for any given \(N\), is shown to be closed under finite direct sums, essential extensions and arbitrary direct products. Necessary and sufficient conditions are derived for this class to be closed under arbitrary direct sums. On the other hand, the class of all modules \(N\) with respect to which an \(R\)-module \(M\) is weakly \(N\)-injective, is proved to be closed under submodules, quotient modules and arbitrary direct sums.
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 1005380
zbMATH Keywords: weak \(N\)-injectivity
zbMATH Keywords: \(N\)-injective modules
zbMATH Keywords: essential extensions
zbMATH Keywords: direct products
reviewed by: Frieda Theron
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1005380