MMN-1693

# On weak Armendariz modules over commutative rings

*M. Shabani*;

*A. Yousefian Darani*;

## Abstract

Let $M$ be a module over commutative a ring $R$. In this paper we generalize some annihilator conditions on rings to modules. Denote by $Nil(M)$ the set of all nilpotent elements of $M$. $M$ is said to be weak Armendariz if $f(x)m(x)=0$, where $f(x)=\sum_{i=0}^n a_ix^i\in R[x]\backslash\{0\}$ and $m(x)=\sum_{j=0}^k m_jx^j\in
M[x]\backslash\{0\}$, then $a_im_j\in Nil(M)$ for each $i=0,1,...,n$ and $j=0,1,...,k$. We prove that the class of these modules are closed under direct sum, finite product and localization. We prove that if $\frac{M}{N}$ is weak Armendariz, then so is $M$.
Furthermore, we show that if $D$-module $M$ is torsion, for a domain $D$, then $M$ is weak Armendariz if and only if $T(M)$ is weak Armendariz,where $T(M)$ is the torsion submodule of $M$.

Vol. 19 (2018), No. 1, pp. 581-590