Closure operators in modules and their characterizations

This work is dedicated to the investigation of closure operators of a module category $R$-Mod. The principal types of closure operators of $R$-Mod are studied and their characterizations are indicated, using dense or (and) closed submodules. The method of description of the closure operators consists in the elucidation of properties of functions which separate in every module, the set of dense submodules and the set of closed submodules. The main properties of the closure operators of $R$-Mod are studied: weakly heredity – idempotency, maximality – minimality, heredity – coheredity, as well as diverse combinations of them. Altogether, 16 types of the closure operators are described, among which 7 types possess double characterizations (by dense submodules and by closed ones).