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Journal ArticleDOI

On Baer and quasi-Baer rings

01 Mar 1970-Duke Mathematical Journal (Duke University Press)-Vol. 37, Iss: 1, pp 127-138
About: This article is published in Duke Mathematical Journal.The article was published on 1970-03-01. It has received 98 citations till now. The article focuses on the topics: Baer ring.
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Journal ArticleDOI
TL;DR: In this article, the authors investigated the relation between Baer rings and p.p.-rings and showed that the skew power series ring R[[x;α]] is a Baer ring if and only if the skew-power series ring π is a p-rigid ring.

194 citations

Journal ArticleDOI
TL;DR: In this article, the set of annihilators in a polynomial ring over a ring was studied, and the relation between the annihilators and the set in R and R was studied.

183 citations

Journal ArticleDOI
TL;DR: In this paper, a ring with unity is right principally quasi-Baer if the right annihilator of a principal right ideal is generated (as a right ideal) by an idempotent.
Abstract: We say a ring with unity is right principally quasi-Baer (or simply, right pq-Baer) if the right annihilator of a principal right ideal is generated (as a right ideal) by an idempotent This class of rings includes the biregular rings and is closed under direct products and Morita invariance The 2-by-2 formal upper triangular matrix rings of this class are characterized Connections to related classes of rings (eg, right PP, Baer, quasi-Baer, right FPF, right GFC, etc) are investigated Examples to illustrate and delimit the theory are provided

174 citations


Cites background from "On Baer and quasi-Baer rings"

  • ...13) or implicitly in (10), every left PP ring with a complete set of primitive idempotents is quasi-Baer (hence left p....

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Journal ArticleDOI
TL;DR: In this article, the notions of Baer and quasi-Baer properties were introduced in a general module theoretic setting, and it was shown that a module M is Baer if the right annihilator of a (two-sided) left ideal of End(M) is a direct summand of M.
Abstract: We introduce the notions of Baer and quasi-Baer properties in a general module theoretic setting. A module M is called (quasi-) Baer if the right annihilator of a (two-sided) left ideal of End(M) is a direct summand of M. We show that a direct summand of a (quasi-) Baer module inherits the property and every finitely generated abelian group is Baer exactly if it is semisimple or torsion-free. Close connections to the (FI-) extending property are investigated and it is shown that a module M is (quasi-) Baer and (FI-) 𝒦-cononsingular if and only if it is (FI-) extending and (FI-) 𝒦-nonsingular. We prove that an arbitrary direct sum of mutually subisomorphic quasi-Baer modules is quasi-Baer and every free (projective) module over a quasi-Baer ring is a quasi-Baer module. Among other results, we also show that the endomorphism ring of a (quasi-) Baer module is a (quasi-) Baer ring, while the converse is not true in general. Applications of results are provided.

137 citations


Cites background from "On Baer and quasi-Baer rings"

  • ...The n-by-n upper triangular matrix ring over a domain which is not a division ring is a quasi-Baer ring but not Baer (Kaplansky, 1968; Pollingher and Zaks, 1970)....

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  • ...The n-by-n matrix ring over a non-Prüfer commutative domain is also an example of a quasi-Baer ring which is not Baer (Kaplansky, 1968; Pollingher and Zaks, 1970)....

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Journal ArticleDOI
TL;DR: In this article, it was shown that for many polynomial extensions (including formal power series, Laurent polynomials, and Laurent series), a ring R is quasi-Baer if and only if the poynomial extension over R is a quasi-Baer.

131 citations


Cites background from "On Baer and quasi-Baer rings"

  • ...For other terminology see [3] and=or [19]....

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  • ...In [19] Pollingher and Zaks show that the class of quasi-Baer rings is closed under n × n matrix rings and under n × n upper (or lower) triangular matrix rings....

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