When is the annihilating ideal graph of a zero-dimensional quasisemilocal commutative ring complemented?

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• 作者：S. Visweswaran ; ^{s_visweswaran2006@yahoo.co.in" class="auth_mail" title="E-mail the corresponding author} ; Hiren D. Patel
• 关键词：Annihilating ideal graph of a commutative ring ; Complemented graph ; Zero-dimensional quasisemilocal ring ; Special principal ideal ring
• 刊名：Arab Journal of Mathematical Sciences
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：22
• 期：1
• 页码：1-21
• 全文大小：287 K

文摘

Let 1319516614000188&_mathId=si1.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=a6e35170c7d91c305a4c3060782ea8ce" title="Click to view the MathML source">R$R$ be a commutative ring with identity. Let 1319516614000188&_mathId=si2.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=01c755a5e3f2f8fb7bc4fda87dc719f4" title="Click to view the MathML source">A(R)$A\left(R\right)$ denote the collection of all annihilating ideals of 1319516614000188&_mathId=si1.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=a6e35170c7d91c305a4c3060782ea8ce" title="Click to view the MathML source">R$R$ (that is, 1319516614000188&_mathId=si2.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=01c755a5e3f2f8fb7bc4fda87dc719f4" title="Click to view the MathML source">A(R)$A\left(R\right)$ is the collection of all ideals 1319516614000188&_mathId=si5.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=494f2c4f16331baf924683c1c7366069" title="Click to view the MathML source">I$I$ of 1319516614000188&_mathId=si1.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=a6e35170c7d91c305a4c3060782ea8ce" title="Click to view the MathML source">R$R$ which admits a nonzero annihilator in 1319516614000188&_mathId=si1.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=a6e35170c7d91c305a4c3060782ea8ce" title="Click to view the MathML source">R$R$). Let 1319516614000188&_mathId=si8.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=8c05195fad734b494174e4f278d11721" title="Click to view the MathML source">AG(R)$AG\left(R\right)$ denote the annihilating ideal graph of 1319516614000188&_mathId=si1.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=a6e35170c7d91c305a4c3060782ea8ce" title="Click to view the MathML source">R$R$. In this article, necessary and sufficient conditions are determined in order that 1319516614000188&_mathId=si8.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=8c05195fad734b494174e4f278d11721" title="Click to view the MathML source">AG(R)$AG\left(R\right)$ is complemented under the assumption that 1319516614000188&_mathId=si1.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=a6e35170c7d91c305a4c3060782ea8ce" title="Click to view the MathML source">R$R$ is a zero-dimensional quasisemilocal ring which admits at least two nonzero annihilating ideals and as a corollary we determine finite rings 1319516614000188&_mathId=si1.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=a6e35170c7d91c305a4c3060782ea8ce" title="Click to view the MathML source">R$R$ such that 1319516614000188&_mathId=si8.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=8c05195fad734b494174e4f278d11721" title="Click to view the MathML source">AG(R)$AG\left(R\right)$ is complemented under the assumption that 1319516614000188&_mathId=si2.gif&_user=111111111&_pii=S1319516614000188&_rdoc=1&_issn=13195166&md5=01c755a5e3f2f8fb7bc4fda87dc719f4" title="Click to view the MathML source">A(R)$A\left(R\right)$ contains at least two nonzero ideals.