the strongly annihilating-ideal graph of a commutative ring with respect to an ideal

For a commutative ring $r$ with identity, ${rm sag}(r)$ be the graph whose vertices are the nonzero annihilating ideals of $r$ and with two distinct nonzero annihilating ideals $i$ and $j$ joined by an edge when $icap {rm ann}(j)neq (0)$ and $jcap {rm ann}(i) neq (0)$. also, strongly annihilating-ideal graph with respect to an ideal $(i)$, that it is shown by ${rm sag}_i(r)$, is the graph whose vertices are all ideals of $r$ such that $knotsubseteq i$ and for some ideal $j$ that $jnotsubseteq i$, $kj subseteq i$, and distinct vertices $k$ and $j$ are adjacent if and only if $jcap {rm ann}_i(k)notsubseteq i$ and $kcap {rm ann}_i(j)notsubseteq i$. in this paper, we study the notion of ${rm sag}_i(r)$. also, among other results, we give some results about the relationships between $rm{ sag}_i(r)$ and ${rm sag}(r/i)$.