# Annihilator-small submodules

Document Type : Research Paper

Authors

1 Mazandaran University, Department of mathematic

2 Hacettepe University, Mathematics Department

Abstract

Let $M_R$ be a module with $S=End(M_R)$. We call a submodule $K$ of
$M_R$ annihilator-small if $K+T=M$, $T$ a submodule of $M_R$,
implies that $ell_S(T)=0$, where $ell_S$ indicates the left
annihilator of $T$ over $S$. The sum $A_R(M)$ of all such submodules
of $M_R$ contains the Jacobson radical $Rad(M)$ and the left
singular submodule $Z_S(M)$. If $M_R$ is cyclic, then $A_R(M)$ is
the unique largest annihilator-small submodule of $M_R$. We study
$A_R(M)$ and $K_S(M)$ in this paper. Conditions when $A_R(M)$ is
annihilator-small and $K_S(M)=J(S)=Tot(M, M)$ are given.

Keywords

Main Subjects

### History

• Receive Date: 02 December 2011
• Revise Date: 24 September 2012
• Accept Date: 25 September 2012
• First Publish Date: 15 December 2013