On \(\xi \)-torsion modules (Q1674238)

For a ring \(R\), a right \(R\)-module \(M\) and a set of right ideals \(\xi\) of \(R\), the subset \(\mathcal{T} _\xi (M)\) of \(M\) is defined as the set of all \(m\) in \(M\) such that \(mI=0\) for some nonzero \(I\) in \(\xi\). If \(\mathcal{T} _\xi (M)=0\), then \(M\) is called \(\xi\)-torsion-free and if \(\mathcal{T} _\xi (M)=M\), then \(M\) is said to be \(\xi\)-torsion. It is shown that every \(\xi\)-torsion module is torsion, but the converse is not true. Closure of the class of \(\xi\)-torsion modules under homomorphic images, submodules and direct sums is investigated and it is shown that the class of \(\xi\)-torsion-free modules is closed under submodules, direct products and quotient modules. An \(R\)-module \(M\) is said to be (proper) \(\xi\)-torsionable if \(\mathcal{T} _\xi (M)\) is a (proper) submodule of \(M\). Some simple examples of \(\xi\)-torsionable modules are given and some properties are explored.