On a theorem by Brewer

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• 作者：Le Thi Ngoc Giau ^{ngocgiau@postech.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Byung Gyun Kang ; ^{bgkang@postech.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：13A15 ; 13C15 ; 13F25
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：221
• 期：1
• 页码：36-44
• 全文大小：318 K

文摘

One of the most frequently referenced monographs on power series rings, “Power Series over Commutative Rings” by James W. Brewer, states in Theorem 21 that if M is a non-SFT maximal ideal of a commutative ring R   with identity, then there exists an infinite ascending chain of prime ideals in the power series ring class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300573&_mathId=si1.gif&_user=111111111&_pii=S0022404916300573&_rdoc=1&_issn=00224049&md5=747592f49fafc69ba050646c45e341f4" title="Click to view the MathML source">R〚X〛class="mathContainer hidden">class="mathCode">$R〚X〛$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300573&_mathId=si19.gif&_user=111111111&_pii=S0022404916300573&_rdoc=1&_issn=00224049&md5=deae601c6b157957629b804a1063e4f7" title="Click to view the MathML source">Q₀⊊Q₁⊊⋯⊊Q_n⊊⋯class="mathContainer hidden">class="mathCode">$Q_0⊊Q_1⊊\cdots ⊊Q_n⊊\cdots$ such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300573&_mathId=si3.gif&_user=111111111&_pii=S0022404916300573&_rdoc=1&_issn=00224049&md5=c4d9fd1f98c614973fb6fcf092d3aa78" title="Click to view the MathML source">Q_n∩R=Mclass="mathContainer hidden">class="mathCode">$Q_n\cap R=M$ for each n  . Moreover, the height of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300573&_mathId=si4.gif&_user=111111111&_pii=S0022404916300573&_rdoc=1&_issn=00224049&md5=ca404f765ab8fd5db7afe94c878f0369" title="Click to view the MathML source">M〚X〛class="mathContainer hidden">class="mathCode">$M〚X〛$ is infinite. In this paper, we show that the above theorem is false by presenting two counter examples. The first counter example shows that the height of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300573&_mathId=si4.gif&_user=111111111&_pii=S0022404916300573&_rdoc=1&_issn=00224049&md5=ca404f765ab8fd5db7afe94c878f0369" title="Click to view the MathML source">M〚X〛class="mathContainer hidden">class="mathCode">$M〚X〛$ can be zero (and hence there is no chain class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300573&_mathId=si19.gif&_user=111111111&_pii=S0022404916300573&_rdoc=1&_issn=00224049&md5=deae601c6b157957629b804a1063e4f7" title="Click to view the MathML source">Q₀⊊Q₁⊊⋯⊊Q_n⊊⋯class="mathContainer hidden">class="mathCode">$Q_0⊊Q_1⊊\cdots ⊊Q_n⊊\cdots$ of prime ideals in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300573&_mathId=si1.gif&_user=111111111&_pii=S0022404916300573&_rdoc=1&_issn=00224049&md5=747592f49fafc69ba050646c45e341f4" title="Click to view the MathML source">R〚X〛class="mathContainer hidden">class="mathCode">$R〚X〛$ satisfying class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300573&_mathId=si3.gif&_user=111111111&_pii=S0022404916300573&_rdoc=1&_issn=00224049&md5=c4d9fd1f98c614973fb6fcf092d3aa78" title="Click to view the MathML source">Q_n∩R=Mclass="mathContainer hidden">class="mathCode">$Q_n\cap R=M$ for each n). In this example, the ring R   is one-dimensional. In the second counter example, we prove that even if the height of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300573&_mathId=si4.gif&_user=111111111&_pii=S0022404916300573&_rdoc=1&_issn=00224049&md5=ca404f765ab8fd5db7afe94c878f0369" title="Click to view the MathML source">M〚X〛class="mathContainer hidden">class="mathCode">$M〚X〛$ is uncountably infinite, there may be no infinite chain class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300573&_mathId=si6.gif&_user=111111111&_pii=S0022404916300573&_rdoc=1&_issn=00224049&md5=867a7c74938c2da292f5f07b0fcb414c" title="Click to view the MathML source">{Q_n}class="mathContainer hidden">class="mathCode">$\{Q_n\}$ of prime ideals in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300573&_mathId=si1.gif&_user=111111111&_pii=S0022404916300573&_rdoc=1&_issn=00224049&md5=747592f49fafc69ba050646c45e341f4" title="Click to view the MathML source">R〚X〛class="mathContainer hidden">class="mathCode">$R〚X〛$ satisfying class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300573&_mathId=si3.gif&_user=111111111&_pii=S0022404916300573&_rdoc=1&_issn=00224049&md5=c4d9fd1f98c614973fb6fcf092d3aa78" title="Click to view the MathML source">Q_n∩R=Mclass="mathContainer hidden">class="mathCode">$Q_n\cap R=M$ for each n.