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Let $R$ be a non-domain commutative ring with identity and $\mathbb{A}^*(R)$ be the set of non-zero ideals with non-zero annihilators‎. ‎We call an ideal $I_1$ of $R$‎, ‎an \textit{annihilating-ideal} if there exists a non-zero ideal $I_2$ of $R$ such that $I_1I_2=(0)$‎. ‎The \textit{annihilating-ideal graph} of $R$ is defined as the graph $\mathbb{AG}(R)$ with the vertex set $\mathbb{A}^*(R)$ and two distinct vertices $I_1$ and $I_2$ are adjacent if and only if $I_1I_2 =(0)$‎. ‎In this paper‎, ‎we characterize all commutative Artinian non-local rings $R$ for which $\mathbb{AG}(R)$ has genus one.