Idempotent Submodules

Abstract

Bican, Jambor, Kepka and Nemec defined a product on the lattice of submodules of a module, making any module into a partially ordered groupoid. Submodules that are idempotent with respect to this product behave similar as idempotent ideals in rings. In particular jansian torsion theories can be described through idempotent submodules. Moreover so-called coclosed submodules, which are essentially closed elements in the dual lattice of submodules of a module, turn out to be idempotent in pi-projective modules. The relation of strongly copolyform modules and the regularity of their endomorphism ring is discussed.