Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph

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• 作者：David F. Anderson^a ; ^{anderson@math.utk.edu} ; John D. LaGrange^b ; ^{lagrangej@lindsey.edu}
• 关键词：13A15 ; 13A99 ; 06E99 ; 05C99 ; 20M14
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2012
• 出版时间：July, 2012
• 年：2012
• 卷：216
• 期：7
• 页码：1626-1636
• 全文大小：298 K

文摘

Let be a commutative ring with . The zero-divisor graph of is the (undirected) graph whose vertices consist of the nonzero zero-divisors of such that distinct vertices and are adjacent if and only if . The relation on given by if and only if is an equivalence relation. The compressed zero-divisor graph is the (undirected) graph whose vertices are the equivalence classes induced by other than and , such that distinct vertices and are adjacent in if and only if . We investigate when is reduced and are interested in when for a reduced ring . Among other results, it is shown that for some Boolean ring if and only if (and hence ) is a complemented graph, and this is equivalent to the total quotient ring of being a von Neumann regular ring.