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    The exact annihilating-ideal graph of a commutative ring

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    The rings considered in this article are commutative with identity. For an ideal II of a ring RR, we denote the annihilator of II in RR by Ann(I)Ann(I). An ideal II of a ring RR is said to be an exact annihilating ideal if there exists a non-zero ideal JJ of RR such that Ann(I)=JAnn(I) = J and Ann(J)=IAnn(J) = I. For a ring RR, we denote the set of all exact annihilating ideals of RR by EA(R)\mathbb{EA}(R) and EA(R)\{(0)}\mathbb{EA}(R)\backslash \{(0)\} by EA(R)βˆ—\mathbb{EA}(R)^{*}. Let RR be a ring such that EA(R)βˆ—β‰ βˆ…\mathbb{EA}(R)^{*}\neq \emptyset. With RR, in [Exact Annihilating-ideal graph of commutative rings, {\it J. Algebra and Related Topics} {\bf 5}(1) (2017) 27-33] P.T. Lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of RR, denoted by EAG(R)\mathbb{EAG}(R) whose vertex set is EA(R)βˆ—\mathbb{EA}(R)^{*} and distinct vertices II and JJ are adjacent if and only if Ann(I)=JAnn(I) = J and Ann(J)=IAnn(J) = I. In this article, we continue the study of the exact annihilating-ideal graph of a ring. In Section 2 , we prove some basic properties of exact annihilating ideals of a commutative ring and we provide several examples. In Section 3, we determine the structure of EAG(R)\mathbb{EAG}(R), where either RR is a special principal ideal ring or RR is a reduced ring which admits only a finite number of minimal prime ideals

    The exact annihilating-ideal graph of a commutative ring

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    The rings considered in this article are commutative with identity. For an ideal II of a ring RR, we denote the annihilator of II in RR by Ann(I)Ann(I). An ideal II of a ring RR is said to be an exact annihilating ideal if there exists a non-zero ideal JJ of RR such that Ann(I)=JAnn(I) = J and Ann(J)=IAnn(J) = I. For a ring RR, we denote the set of all exact annihilating ideals of RR by EA(R)\mathbb{EA}(R) and EA(R)\{(0)}\mathbb{EA}(R)\backslash \{(0)\} by EA(R)βˆ—\mathbb{EA}(R)^{*}. Let RR be a ring such that EA(R)βˆ—β‰ βˆ…\mathbb{EA}(R)^{*}\neq \emptyset. With RR, in [Exact Annihilating-ideal graph of commutative rings, {\it J. Algebra and Related Topics} {\bf 5}(1) (2017) 27-33] P.T. Lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of RR, denoted by EAG(R)\mathbb{EAG}(R) whose vertex set is EA(R)βˆ—\mathbb{EA}(R)^{*} and distinct vertices II and JJ are adjacent if and only if Ann(I)=JAnn(I) = J and Ann(J)=IAnn(J) = I. In this article, we continue the study of the exact annihilating-ideal graph of a ring. In Section 2 , we prove some basic properties of exact annihilating ideals of a commutative ring and we provide several examples. In Section 3, we determine the structure of EAG(R)\mathbb{EAG}(R), where either RR is a special principal ideal ring or RR is a reduced ring which admits only a finite number of minimal prime ideals
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