Taheri, R., Tehranian, A. (2016). The principal ideal subgraph of the annihilating-ideal graph of commutative rings. Algebraic Structures and Their Applications, 3(1), 39-52.

Reza Taheri; Abolfazl Tehranian. "The principal ideal subgraph of the annihilating-ideal graph of commutative rings". Algebraic Structures and Their Applications, 3, 1, 2016, 39-52.

Taheri, R., Tehranian, A. (2016). 'The principal ideal subgraph of the annihilating-ideal graph of commutative rings', Algebraic Structures and Their Applications, 3(1), pp. 39-52.

Taheri, R., Tehranian, A. The principal ideal subgraph of the annihilating-ideal graph of commutative rings. Algebraic Structures and Their Applications, 2016; 3(1): 39-52.

The principal ideal subgraph of the annihilating-ideal graph of commutative rings

^{}Islamic Azad University, Science and Research Branch, Tehran, Iran

Abstract

Let $R$ be a commutative ring with identity and $\mathbb{A}(R)$ be the set of ideals of $R$ with non-zero annihilators. In this paper, we first introduce and investigate the principal ideal subgraph of the annihilating-ideal graph of $R$, denoted by $\mathbb{AG}_P(R)$. It is a (undirected) graph with vertices $\mathbb{A}_P(R)=\mathbb{A}(R)\cap \mathbb{P}(R)\setminus \{(0)\}$, where $\mathbb{P}(R)$ is the set of proper principal ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. Then, we study some basic properties of $\mathbb{AG}_P(R)$. For instance, we characterize rings for which $\mathbb{AG}_P(R)$ is finite graph, complete graph, bipartite graph or star graph. Also, we study diameter and girth of $\mathbb{AG}_P(R)$. Finally, we compare the principal ideal subgraph $\mathbb{AG}_P(R)$ and spectrum subgraph $\mathbb{AG}_s(R)$.

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