Decompositions of Baer-like rings

Birkenmeier, Gary F.

doi:10.1007/bf00050894

Cited by 32 publications

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“…Annihilators and annihilator ideals have been studied in different classes of algebras. In ring theory, Bear rings, quasi-Bear rings, and principally quasi-Baer rings are defined using left and right annihilator ideals (see [1][2][3][4][5][6]). e notion of annihilators is also closely connected to that of regular rings, specifically to biregular rings, Von Neumann regular rings, self-injective rings, and continuous regular rings (see [7][8][9][10][11][12]).…”

Section: Introductionmentioning

confidence: 99%

Annihilators in Universal Algebras: A New Approach

Addis

2020

Journal of Mathematics

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Section: Introductionmentioning

confidence: 99%

Annihilators in Universal Algebras: A New Approach

Addis

2020

Journal of Mathematics

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“…Recall that R is (quasi-) Baer if the right annihilator of every nonempty subset (every right ideal) of R is generated by an idempotent. A lot of works on Baer rings and quasi-Baer rings appears in [3][4][5][6]9]. As a generalization of quasi-Baer rings, G.F. Birkenmeier, J.Y.…”

Section: Introductionmentioning

confidence: 99%

Principal Quasi-Baerness of Modules of Generalized Power Series

Zhao¹,

Jiao²

2011

Taiwanese J. Math.

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Let R be a ring, M a right R-module and (S, ≤) a strictly totally ordered monoid. It is shown that [[M S,≤ ]], the module of generalized power series with coefficients in M and exponents in S, is a p.q.Baer right [[R S,≤ ]]module if and only if the right annihilator of any S-indexed family of cyclic submodules of M in R is generated by an idempotent of R. Furthermore, we will show that for a ring R with all left semicentral idempotents are central, the ring [[R S,≤ ]] consisting of generalized power series over R is a right p.q.Baer ring if and only if R is a right p.q.Baer ring and any S-indexed family of central idempotents of R has a generalized join in I(R), where I(R) is the set of all idempotents of R.

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“…Baer rings were studied in [1,2,3,5,6,7,11]. By [5, Theorem 3] the Baer condition is left-right symmetric.…”

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Baer rings of generalized power series Research supported by the NSF of China (10171082) and by the Foundation for University Key Teachers of the Ministry of Education.

Liu¹

2002

Glasgow Math. J.

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Abstract. We show that if R is a commutative ring and ðS; Þ a strictly totally ordered monoid, then the ring ½½R S; of generalized power series is Baer if and only if R is Baer.2000 Mathematics Subject Classification. 13F25, 16W60.A ring R is called Baer if the right annihilator of every nonempty subset of R is generated by an idempotent. Baer rings were studied in [1,2,3,5,6,7,11]. By [5, Theorem 3] the Baer condition is left-right symmetric. Semisimple artinian rings, domains and the rings of n Â n upper triangular matrices over division rings are Baer, where n ¼ 1; 2; . . ..A ring R is called a right pp-ring if each principal right ideal of R is projective, or equivalently, if the right annihilator of each element of R is generated by an idempotent. Baer rings are clearly right pp-rings. It was proved in [9] that if R is a commutative ring and ðS; Þ a strictly totally ordered monoid then the ring ½½R S; of generalized power series is a pp-ring if and only if R is a pp-ring and every S-indexed subset C of the set BðRÞ of all idempotents of R has a least upper bound in BðRÞ. In this paper we show that if R is a commutative ring and ðS; Þ a strictly totally ordered monoid, then the ring ½½R S; of generalized power series is Baer if and only if R is Baer. All rings considered here are associative with identity. Any concept and notation not defined here can be found in [12,13,14,15]. Let ðS; Þ be an ordered set. Recall that ðS; Þ is artinian if every strictly decreasing sequence of elements of S is finite, and that ðS; Þ is narrow if every subset of pairwise order-incomparable elements of S is finite. Let S be a commutative monoid. Unless stated otherwise, the operation of S will be denoted additively, and the neutral element by 0. The following definition is due to P. Ribenboim. See [12,13,14,15].Let ðS; Þ be a strictly ordered monoid (that is, ðS; Þ is an ordered monoid satisfying the condition that, if s; s 0 ; t 2 S and s < s 0 , then s þ t < s 0 þ t), and R a commutative ring. Let A ¼ ½½R S; be the set of all maps f : SÀ!R such that suppð f Þ ¼ fs 2 Sj fðsÞ 6 ¼ 0g is artinian and narrow. With pointwise addition, A is an abelian additive group. For every s 2 S and f; g 2 A, let X s ð f; gÞ ¼ fðu; vÞ 2 S Â Sj s ¼ u þ v; fðuÞ 6 ¼ 0; gðvÞ 6 ¼ 0g. It follows from [14, 1.16] that X s ð f; gÞ is finite. This fact allows us to define the operation of convolution

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Decompositions of Baer-like rings

Cited by 32 publications

References 20 publications

Annihilators in Universal Algebras: A New Approach

Annihilators in Universal Algebras: A New Approach

Principal Quasi-Baerness of Modules of Generalized Power Series

Baer rings of generalized power series Research supported by the NSF of China (10171082) and by the Foundation for University Key Teachers of the Ministry of Education.

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