Polynomial extensions of Baer and quasi-Baer rings

Birkenmeier, Gary F.; Kim, Jin Yong; Park, Jae Keol

doi:10.1016/s0022-4049(00)00055-4

Cited by 115 publications

(46 citation statements)

References 12 publications

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“…Right p.q.-Baer rings have been initially studied in [1]. For more details on (right) p.q.-Baer rings, see [1][2][3][4][5][6]. A ring R is called quasi-Baer if the right annihilator of every right ideal is generated, as a right ideal by an idempotent of R in [7] (see also [8].…”

Section:  mentioning

confidence: 99%

The p.q.-Baer Property of Skew Group Rings under Finite Group Action*

Li¹,

Jin²

2013

APM

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Section:  mentioning

confidence: 99%

The p.q.-Baer Property of Skew Group Rings under Finite Group Action*

Li¹,

Jin²

2013

APM

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“…Right p.q.-Baer rings have been initially studied in [1]. For more details on (right) p.q.-Baer rings, see [1][2][3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

The p.q.-Baer Property of Fixed Rings under Finite Group Action

Liu¹,

Jin²

2012

APM

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“…Birkenmeier et al in [5,Theorem 1.2] show that if R is a quasi-Baer ring and α is an automorphism of R, then R[x; α] is a quasi-Baer ring. They also provided an example of a quasi-Baer ring R with an endomorphism α such that R[x; α] is not quasi-Baer.…”

mentioning

confidence: 99%

“…They also provided an example of a quasi-Baer ring R with an endomorphism α such that R[x; α] is not quasi-Baer. In [7], they also proved that a ring R is right p.q.-Baer if and only if the polynomial ring R[x] is right p.q.-Baer.…”

mentioning

confidence: 99%

On Weakly Rigid Rings

Nasr-Isfahani

Moussavi

2009

Glasgow Math. J.

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Abstract. Let R be a ring with a monomorphism α and an α-derivation δ. We introduce (α, δ)-weakly rigid rings which are a generalisation of α-rigid rings and investigate their properties. Every prime ring R is (α, δ)-weakly rigid for any automorphism α and α-derivation δ. It is proved that for any n, a ring R is (α, δ)-weakly rigid if and only if the n-by-n upper triangular matrix ring T n (R) is (ᾱ,δ)-weakly rigid if and only if M n (R) is (ᾱ,δ)-weakly rigid. Moreover, various classes of (α, δ)-weakly rigid rings is constructed, and several known results are extended. We show that for an (α, δ)-weakly rigid ring R, and the extensions, the ring R is quasi-Baer if and only if the extension over R is quasi-Baer. It is also proved that for an (α, δ)-weakly rigid ring R, if any one of the rings R,is left principally quasi-Baer, then so are the other three. Examples to illustrate and delimit the theory are provided.AMS Subject Classification. 16S36. 16W60.1. Introduction. Throughout this paper R denotes an associative ring with unity; α is a monomorphism of R which is not assumed to be surjective; and δ an α-derivation of R, that is δ is an additive map such thatAccording to Krempa [18], a monomorphism α of a ring R is called to be rigid if aα(a) = 0 implies a = 0 for a ∈ R. A ring R is said to be α-rigid if there exists a rigid monomorphism α of R.The By [12], a ring R is α-rigid if and only if it is (α, δ)-compatible and reduced. Notice that the class of α-rigid rings and (α, δ)-compatible rings is a narrow class of rings, and it is easy to see that every (α, δ)-compatible ring is (α, δ)-weakly rigid; but there are various classes of (α, δ)-weakly rigid rings which are not (α, δ)-compatible, as we will see in Section 2.It is clear that every prime ring R is (α, δ)-weakly rigid for any automorphism α and α-derivation δ. In this paper we prove that for any positive integer n, a ring R is (α, δ)-weakly rigid if and only if the n-by-n upper triangular matrix ring T n (R) is (ᾱ,δ)-weakly rigid if and only if the matrix ring M n (R) is (ᾱ,δ)-weakly rigid.

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Polynomial extensions of Baer and quasi-Baer rings

Cited by 115 publications

References 12 publications

The p.q.-Baer Property of Skew Group Rings under Finite Group Action*

The p.q.-Baer Property of Skew Group Rings under Finite Group Action*

The p.q.-Baer Property of Fixed Rings under Finite Group Action

On Weakly Rigid Rings

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