On cocomposition series relative to a module
Let \(R\) be an associative ring with identity, Mod-\(R\) the category of all unital right \(R\)-modules, and \(\pi:\text{Mod-}R\to\text{Mod-}R\); \(M\mapsto\pi(M)=\sum\{\text{Im }f\mid f\in\text{Hom}_R(P,M)\}\) an idempotent pretorsion functor on Mod-\(R\), where \(P\in\text{Mod-}R\). An \(R\)-module \(M\) is called \(P\)-cotorsionless if \(\pi(M)=M\), and is called \(P\)-cotorsion if \(\pi(M)=0\). A nonzero right \(R\)-module \(V\) is called \(P\)-neat if \(V\) is \(P\)-cotorsionless and any proper submodule of \(V\) is \(P\)-cotorsion. A chain of \(R\)-submodules of \(M \in \text{Mod-}R\) from 0 to \(\pi(M)\) \(0=L_0<L_1<\dots<L_n=\pi(M)\) is called a \(P\)-cocomposition series of \(M\) if the factor \(L_i/L_{i-1}\) is \(P\)-neat for \(i=1,2,\dots,n\). The author gives a method to construct \(P\)-cocomposition series of \(N\) and \(M/N\) from a \(P\)-cocomposition series of \(M\) and, on the other hand, to construct a \(P\)-cocomposition series of \(M\) from \(P\)-cocomposition series of \(N\) and \(M/N\). Moreover, from this follows immediately Theorem 2.11 in a paper by \textit{T. Izawa} [J. Algebra 118, No. 2, 388-407 (1988; Zbl 0659.16019)].
scientific article; zbMATH DE number 853031