Nil-quasipolar rings

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• 作者：Orhan Gurgun ; Sait Halicioglu…
• 关键词：Nil ; quasipolar matrix ; Quasipolar ring ; Strongly nil ; clean ring ; Matrix ring ; Characteristic polynomial ; 16S50 ; 16S70 ; 16U99
• 刊名：Bolet篓陋n de la Sociedad Matem篓垄tica Mexicana
• 出版年：2014
• 出版时间：April 2014
• 年：2014
• 卷：20
• 期：1
• 页码：29-38
• 全文大小：213 KB
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• 作者单位：Orhan Gurgun (1)
  Sait Halicioglu (1)
  Abdullah Harmanci (2)
  
  1. Department of Mathematics, Ankara University, Ankara, Turkey
  2. Department of Maths, Hacettepe University, Ankara, Turkey
  
• 刊物类别：Mathematics, general;
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：2296-4495

文摘

Let \(R\) be an arbitrary ring. An element \(a\in R\) is nil-quasipolar if there exists \(p^2=p\in comm^2(a)\) such that \(a+p\in Nil(R)\); \(R\) is called nil-quasipolar in case each of its elements is nil-quasipolar. In this paper, we study nil-quasipolar rings over commutative local rings. We determine the conditions under which a single \(2\times 2\) matrix over a commutative local ring is nil-quasipolar. It is shown that \(A\in M_2(R)\) is nil-quasipolar if and only if \(A\in Nil\big (M_2(R)\big )\) or \(A+I_2\in Nil\big (M_2(R)\big )\) or the characteristic polynomial \(\chi (A)\) has a root in \(Nil(R)\) and a root in \(-1+Nil(R)\). We give some equivalent characterizations of nil-quasipolar rings through the endomorphism ring of a module. Among others we prove that every nil-quasipolar ring has stable range one. Keywords Nil-quasipolar matrix Quasipolar ring Strongly nil-clean ring Matrix ring Characteristic polynomial