ON THE RELATIVE FINITE DIMENSION IN MODULE OVER RINGS

We consider an R-module M where R is an associative ring. Uniform submodules play a significant role to establish various finite dimension conditions in modules over associative rings. However, certain results can not be generalized to modules over rings unless we impose some assumption(s). In particular, we obtain the relative finite Goldie dimension of the sum of two submodules of a module over a ring if their intersection is a complement submodule. We consider the notions such as relative essential submodule, relative uniform submodule of an R-module M, and prove various finite dimensional conditions of submodules of M. We provide suitable examples distinguishing the existing notions.