On fissible modules (Q1332966)

The author considers the additive groups of modules. Let \(R\) be an associative ring with unit, and \(M\) be a unital left \(R\)-module. A ring \(R\) is fissible if \(R\) is a direct sum \(R = R_ t \oplus S\) (\(R_ t\) is the torsion part of \(R^ +\)). If \(R\) is a fissible ring, then \(R_ t\) is bounded. An \(R\)-module \(M\) is fissible if \(M\) is an \(R\)-module direct sum \(M = M_ t \oplus N\). Every \(R\)-module is fissible if and only if \(R\) is a ring direct sum \(R = R_ t \oplus S\) such that every \(S\)-module is fissible. Theorem 1. Every \(R\)-module is fissible if and only if \(R\) is a ring direct sum \(R = R_ t \oplus D\) with \(D^ +\) a torsion-free divisible group. An \(R\)-module \(M\) is \(p\)-fissible, \(p\) a prime, if \(M\) is an \(R\)-module direct sum \(M = M_ p \oplus N\) (\(M_ p\) is the \(p\)- primary component of \(M^ +\)). Theorem 2. Let \(p\) be a prime. Every \(R\)- module is \(p\)-fissible if and only if \(R\) is a direct sum \(R = R_ p \oplus S\) with \(S^ +\) a \(p\)-divisible group.