Generalised reduced modules
An \(R\)-module \(M\) is reduced if for all \(a\in R\) and \(m\in M,a^{2}m=0\) implies that \(am=0\). An equivalent formulation of a reduced module is in terms of the \(a\)-torsion functor \(\Gamma _{a}\) which assigns to an \(R\)-module \(M\) a submodule \(\Gamma _{a}(M)=\{m\in M\mid a^{k}m=0\) for some \(k\in\mathbb{Z}^{+}\}\). Then \(M\) is reduced iff \(a\Gamma _{a}(M)=0\) for all \(a\in R.\) Here the authors present a generalisation of this notion. For a positive integer \(t\) and \(a\in R\), an \(R\)-module \(M\) is \(a^{t}\)\textit{-reduced} if for every positive integer \(k\geq t\) and \(m\in M\), \(a^{k}m=0\) implies that \(a^{t}m=0\) (or equivalently, \(a^{t}\Gamma _{a}(M)=0)\) and \(M\) is \textit{universally} \(a^{t}\)\textit{-reduced} if it is \(a^{t}\)-reduced for all \(a\in R\). A number of characterizations and properties of these notions are given and the relationship between such modules and the corresponding ring is clarified.
scientific article; zbMATH DE number 7658704
scientific article