Uniformly S-essential submodules and uniformly S-injective uniformly S-envelopes
Mohammad Adarbeh, Mohammad Saleh
TL;DR
The paper introduces uniformly $S$-essential ($u$-$S$-essential) submodules and uniformly $S$-injective envelopes in the setting of a commutative ring $R$ with a multiplicative subset $S$, and defines associated notions such as $u$-$S$-uniform modules and $u$-$S$-torsion. It develops a framework in which $u$-$S$-injective $u$-$S$-envelopes are characterized by $u$-$S$-essential submodules, and proves that finite direct sums of such envelopes retain the enveloping property whereas infinite sums need not. The work establishes key equivalences and stability properties, including when $u$-$S$-essential submodules coincide with essential submodules (e.g., $S$-torsion-free modules or $S\subseteq U(R)$) and the role of primes and maximal ideals in the theory. It concludes with examples, general constructions, and an open question on the universal existence of $u$-$S$-injective $u$-$S$-envelopes for all modules.
Abstract
In this paper, we introduce the notion of uniformly S-essential (u-S-essential) submodules. Let R be a commutative ring and S a multiplicative subset of R. A submodule K of an R-module M is said to be u-S-essential in M if for any submodule L of M, s1(K \cap L) = 0 for some s1 \in S implies s2L = 0 for some s2 \in S. Several properties of this notion are studied. The notions of a u-S-uniform module and a u-S-injective u-S-envelope are also introduced, and we show that these notions are characterized by u-S-essential submodules.