A new class of small submodules

Document Type : Original Manuscript


1 Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran.

2 Department of Mathematics‎, ‎Payame Noor University‎, ‎P.O.Box 19395-3697, ‎Tehran‎, ‎Iran.

3 Department of Mathematics‎, ‎Payame Noor University‎, ‎P.O.Box 19395-3697‎, ‎Tehran‎, ‎Iran



‎Let $R$ be a commutative ring with identity ‎$‎1\neq 0‎$ ‎and $M$ a nonzero unital $R$-module‎. ‎In this paper‎, ‎we introduce a new notion of submodules in $M$‎, ‎namely $T$-semi-annihilator small submodules of $M$ with respect to an arbitrary submodule $T$ of $M‎$‎‎. ‎A submodule $N$ of $M$ is $T$-semi-annihilator small in $M$ provide that for each submodule $X$ of $M$ with‎ ‎$T\subseteq N+X$ implies that ${\rm Ann}(X)\ll (T:M)$‎. ‎In addition‎, ‎we investigate some results concerning to this new class of submodules‎. ‎Among various results‎, ‎we prove that for a faithful finitely generated multiplication module $M$‎, ‎the submodule $N$ of $M$ is a $T$-semi-annihilator small submodule of $M$ if and only if $(N:M)$ is a $(T:M)$-semi-annihilator small ideal of $R$‎. ‎Finally‎, ‎we explore the properties and the behaviour of this structure under ring homomorphism‎, ‎localization‎, ‎direct sums and tensor product of them with a ‎faithfully‎ flat $R$-module‎.