Finitely Generated Annihilating-Ideal Graph of Commutative Rings
سال انتشار: 1397
نوع سند: مقاله ژورنالی
زبان: انگلیسی
مشاهده: 76
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شناسه ملی سند علمی:
JR_IJIM-10-4_006
تاریخ نمایه سازی: 26 دی 1402
چکیده مقاله:
Let R be a commutative ring and mathbb{A}(R) be the set of all ideals with non-zero annihilators. Assume that mathbb{A}^*(R)=mathbb{A}(R)diagdown {۰} and mathbb{F}(R) denote the set of all finitely generated ideals of R. In this paper, we introduce and investigate the {it finitely generated subgraph} of the annihilating-ideal graph of R, denoted by mathbb{AG}_F(R). It is the (undirected) graph with vertices mathbb{A}_F(R)=mathbb{A}^*(R)cap mathbb{F}(R) and two distinct vertices I and J are adjacent if and only if IJ=(۰). First, we study some basic properties of mathbb{AG}_F(R). For instance, it is shown that if R is not a domain, then mathbb{AG}_F(R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, Artinian). We characterize all rings for which mathbb{AG}_F(R) is a finite, complete, star or bipartite graph. Next, we study diameter and girth of mathbb{AG}_F(R). It is proved that {rm diam}(mathbb{AG}_F(R))leqslant {rm diam}(mathbb{AG}(R)) and {rm gr}(mathbb{AG}_F(R))={rm gr}(mathbb{AG}(R)).
کلیدواژه ها:
نویسندگان
R. Taheri
Department of Mathematics, Shahrekord Branch, Islamic Azad Univercsity, Shahrekord, Iran.
A. Tehranien
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran