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The sum-annihilating essential ideal graph of a commutative ring | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 3، دوره 1، شماره 2، اسفند 2016، صفحه 117-135 اصل مقاله (553.67 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2016.13555 | ||
| نویسندگان | ||
| Abbas Alilou؛ Jafar Amjadi* | ||
| Azarbaijan Shahid Madani University | ||
| چکیده | ||
| Let $R$ be a commutative ring with identity. An ideal $I$ of a ring $R$ is called an annihilating ideal if there exists $r\in R\setminus \{0\}$ such that $Ir=(0)$ and an ideal $I$ of $R$ is called an essential ideal if $I$ has non-zero intersection with every other non-zero ideal of $R$. The sum-annihilating essential ideal graph of $R$, denoted by $\mathcal{AE}_R$, is a graph whose vertex set is the set of all non-zero annihilating ideals and two vertices $I$ and $J$ are adjacent whenever ${\rm Ann}(I)+{\rm Ann}(J)$ is an essential ideal. In this paper we initiate the study of the sum-annihilating essential ideal graph. We first characterize all rings whose sum-annihilating essential ideal graph are stars or complete graphs and then establish sharp bounds on domination number of this graph. Furthermore determine all isomorphism classes of Artinian rings whose sum-annihilating essential ideal graph has genus zero or one. | ||
| کلیدواژه‌ها | ||
| Commutative rings؛ annihilating ideal؛ essential ideal؛ genus of a graph | ||
| مراجع | ||
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