MODULES WHOSE t-CLOSED SUBMODULES HAVE A SUMMAND AS A COMPLEMENT

摘要

We define and investigate T_(11)-type modules as a generalization of t-extending modules, and modules satisfying C_(11) condition. A module M is said to be T_(11)-type if every t-closed submodule of M has a complement which is a direct summand. Direct sums of T_(11)-type modules inherit the property. Some equivalent conditions for a module M to be T_(11)-type are given. We characterize a module M for which every direct summand satisfies T_(11) condition. If R_R is T_(11)-type, then R/Z_2(R_R) is a C_2 ring if and only if it is a von Neumann regular ring. Applying this result, we characterize a right t-extending (resp., finitely ∑-t-extending, or ∑-t-extending) ring R for which R/Z_2(R_R) is von Neumann regular.