The annihilating-submodule graph of modules over commutative rings II

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• 作者：Habibollah Ansari-Toroghy ; Shokoufeh Habibi
• 关键词：Mathematics Subject Classification05C75 ; 13C13
• 刊名：Arabian Journal of Mathematics
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：5
• 期：4
• 页码：187-194
• 全文大小：505 KB
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：2193-5351
• 卷排序：5

文摘

Let M be a module over a commutative ring R. The annihilating-submodule graph of M, denoted by AG(M), is a simple graph in which a non-zero submodule N of M is a vertex if and only if there exists a non-zero proper submodule K of M such that N K = (0), where N K, the product of N and K, is denoted by (N : M)(K : M)M and two distinct vertices N and K are adjacent if and only if N K = (0). This graph is a submodule version of the annihilating-ideal graph. We prove that if AG(M) is a tree, then either AG(M) is a star graph or a path of order 4 and in the latter case \({M\cong F \times S}\), where F is a simple module and S is a module with a unique non-trivial submodule. Moreover, we prove that if M is a cyclic module with at least three minimal prime submodules, then gr(AG(M)) = 3 and for every cyclic module M, \({cl({\rm AG}(M)) \geq |{\rm Min}(M)|}\).