Uniformly S-essential submodules and uniformly S-injective uniformly S-envelopes

In this paper, we introduce the notion of uniformly S-essential (u-S-essential) submodules. Let R be a commutative ring, S a multiplicative subset of R, and M an R-module. A submodule N of M is said to be u-S-essential in M if for any submodule L of M, N \cap L is u-S-torsion implies L is u-S-torsion. Several properties of this notion are studied. We also introduce the notions of u-S-uniform modules and u-S-injective u-S-envelopes and characterize them in terms of u-S-essential submodules.