Let $M$ be a module over a commutative ring $R$, $Z_{*}(M)$ be its set of weak zero-divisor elements, and if $m\in M$, then let $I_m=(Rm:_R M)=\{r\in R : rM\subseteq Rm\}$. The annihilator graph of $M$ is the (undirected) graph $AG(M)$ with vertices $\tilde{Z_{*}}(M)=Z_{*}(M)\setminus \{0\}$, and two distinct vertices $m$ and $n$ are adjacent if and only if $(0:_R I_{m}I_{n}M)\neq (0:_R m)\cup (0:_R n)$. We show that $AG(M)$ is connected with diameter at most two and girth at most four. Also, we study some properties of the zero-divisor graph of reduced multiplication-like $R$-modules.
Esmaeili Khalil Saraei, F. (2021). The annihilator graph of modules over commutative rings. Journal of Algebra and Related Topics, 9(1), 93-108. doi: 10.22124/jart.2021.18226.1241
MLA
F. Esmaeili Khalil Saraei. "The annihilator graph of modules over commutative rings". Journal of Algebra and Related Topics, 9, 1, 2021, 93-108. doi: 10.22124/jart.2021.18226.1241
HARVARD
Esmaeili Khalil Saraei, F. (2021). 'The annihilator graph of modules over commutative rings', Journal of Algebra and Related Topics, 9(1), pp. 93-108. doi: 10.22124/jart.2021.18226.1241
VANCOUVER
Esmaeili Khalil Saraei, F. The annihilator graph of modules over commutative rings. Journal of Algebra and Related Topics, 2021; 9(1): 93-108. doi: 10.22124/jart.2021.18226.1241
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