A note on the pure radical of a module
Let \(R\) be a ring with identity and let \(M\) be a nonzero unitary left \(R\)-module. A pure submodule of \(M\) is a submodule \(N\) of \(M\) such that the homomorphism \(L\otimes_R N\to L\otimes_R M\) is injective for every right \(R\)-module \(L\). The author defines the pure radical of a left \(R\)-module \(M\) as the intersection of all maximal pure submodules of \(M\). He then describes the basic characterizations of this radical. For example, he gives a necessary and sufficient condition for the pure radical of a quotient module to be zero. He also establishes a relation between the two notions of the pure radical and the radical of a given \(R\)-module.
scientific article; zbMATH DE number 1092955