Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathscr {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">P</mi> </math></EquationSource> </InlineEquation> be a commutative ring with unity, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A(\mathscr {P})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">P</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denote the set of ideals of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathscr {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">P</mi> </math></EquationSource> </InlineEquation> having a nonzero annihilator. The annihilating-ideal graph of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathscr {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">P</mi> </math></EquationSource> </InlineEquation>, denoted by <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$AG(\mathscr {P})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">P</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, is an undirected graph whose vertex set <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$A(\mathscr {P})^*$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">P</mi> <mo stretchy="false">)</mo> </mrow> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> consists of all nonzero ideals in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$A(\mathscr {P})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">P</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where two distinct vertices I and J are adjacent if and only if <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$IJ=(0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mi>J</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we investigate the structure of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$AG(\mathscr {P})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">P</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for Artinian non-local commutative rings. Our main results provide a complete characterization of all such rings for which the annihilating-ideal graph is a line graph, as well as those for which it is the complement of a line graph. In particular, we show that these graphs arise precisely for rings that decompose into specific finite direct products of fields and local rings with a unique nontrivial ideal.

Line graph of annihilating-ideal graph of commutative rings

期刊论文
发表日期：

2026年3月10日

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Nadeem ur Rehman,
Shabir Ahmad Mir,
Mohd Nazim,
Haribhau R. Bhapkar,
Nazim

摘要

Let $\mathscr {P}$ $P$ be a commutative ring with unity, and let $A(\mathscr {P})$ $A (P)$ denote the set of ideals of $\mathscr {P}$ $P$ having a nonzero annihilator. The annihilating-ideal graph of $\mathscr {P}$ $P$ , denoted by $AG(\mathscr {P})$ $A G (P)$ , is an undirected graph whose vertex set $A(\mathscr {P})^*$ $A {(P)}^{*}$ consists of all nonzero ideals in $A(\mathscr {P})$ $A (P)$ , where two distinct vertices I and J are adjacent if and only if $IJ=(0)$ $I J = (0)$ . In this paper, we investigate the structure of $AG(\mathscr {P})$ $A G (P)$ for Artinian non-local commutative rings. Our main results provide a complete characterization of all such rings for which the annihilating-ideal graph is a line graph, as well as those for which it is the complement of a line graph. In particular, we show that these graphs arise precisely for rings that decompose into specific finite direct products of fields and local rings with a unique nontrivial ideal.