Closure operators in modules and their characterizations.

This work is dedicated to the investigation of closure operators of a mod- ule 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 prop- erties 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 - co- heredity, 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). [ABSTRACT FROM AUTHOR]