A generalization of the zero-divisor graph for modules
| Autor: | Katayoun Nozari, Shiroyeh Payrovi |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Publications de l'Institut Math?matique (Belgrade). 106:39-46 |
| ISSN: | 1820-7405 0350-1302 |
| DOI: | 10.2298/pim1920039n |
| Popis: | Let R be a commutative ring and M a Noetherian R-module. The zero-divisor graph of M, denoted by ?(M), is an undirected simple graph whose vertices are the elements of ZR(M)\AnnR(M) and two distinct vertices a and b are adjacent if and only if abM = 0. In this paper, we study diameter and girth of ?(M). We show that the zero-divisor graph of M has a universal vertex in ZR(M)\r(AnnR(M)) if and only if R = ?Z2?R? and M = Z2?M?, where M? is an R?-module. Moreover, we show that if ?(M) is a complete graph, then one of the following statements is true: (i) AssR(M) = {m1,m2}, where m1,m2 are maximal ideals of R. (ii) AssR(M) = {p}, where p 2 ? AnnR(M). (iii) AssR(M) = {p}, where p 3 ? AnnR(M). |
| Databáze: | OpenAIRE |
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