The lattice of \(P\)-cotorsionless submodules of a module (Q1194190)

Let \(R\) be an associative ring with identity, and \(M\), \(P\) two right \(R\)- modules. Following \textit{T. Izawa} [J. Algebra 118, 388-407 (1988; Zbl 0659.16019)], \(M\) is said to be \(P\)-cotorsionless (resp. \(P\)-cotorsion) if \(M\) is \(P\)-generated (resp. if \(\text{Hom}_ R(P,M) = 0\)). Using these notions, Izawa has introduced and investigated in the above mentioned paper the \(P\)-cocomposition series of \(M\). The aim of the paper under review is to simplify the proofs of some results from Izawa's paper, as well as to obtain new aspects of them. The main tool of the paper is the following result: suppose that \(P\) is \(M\)-projective; then the lattice \({\mathcal S}_ P(M/N)\) of all \(P\)-cotorsionless submodules of \(M/N\) is lattice isomorphic to the interval \([\pi(N),\pi(M)]\) of the lattice \({\mathcal S}_ P(M)\) of all \(P\)-cotorsionless submodules of \(M\), where \(\pi(X)\) denotes for a right \(R\)-module \(X\) the trace of \(P\) in \(X\).