Quasi-Baer ring extensions and biregrular rings

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

doi:10.1017/s0004972700022000

Cited by 37 publications

(11 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

Section:  mentioning

confidence: 99%

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

Li¹,

Jin²

2013

APM

View full text Add to dashboard Cite

“…For this, let E ij denote the matrix in R with 1 in the (i, j)-position and 0 elsewhere. Take f 1 = xE 11 …”

Section: Corollary 39 a Ring R Is Strongly Regular If And Only If Rmentioning

confidence: 99%

“…The classes of Baer, quasi-Baer, right extending, and right FI-extending rings are denoted by B, qB, E, and FI, respectively. (See [8,25,42] for B, [10][11][12]14,21,26,53] for qB, [24,25,30] for E, and [14,18,32] for FI.) The notion I P R means that I is an ideal of a ring R. For a ring R, we use P(R), I(R), B(R), Cen(R), and Mat n (R) to denote the prime radical of R, the set of all idempotents of R, the set of all central idempotents of R, the center of R, and the n-by-n matrix ring over R, respectively.…”

Section: Introductionmentioning

confidence: 99%

Hulls of semiprime rings with applications to C∗-algebras

2009

Self Cite

View full text Add to dashboard Cite

For a ring R, we investigate and determine "minimal" right essential overrings (right ring hulls) belonging to certain classes which are generated by R and subsets of the central idempotents of Q (R), where Q (R) is the maximal right ring of quotients of R. We show the existence of and characterize a quasi-Baer hull and a right FI-extending hull for every semiprime ring. Our results include: (i) RB(Q (R)) (i.e., the subring of Q (R) generated by {re | r ∈ R and e ∈ B(Q (R))}, where B(Q (R)) is the set of all central idempotents of Q (R)) is the smallest quasi-Baer and the smallest right FI-extending right ring of quotients of a semiprime ring R with unity. In this case, various overrings of RB(Q (R)), including all right essential overrings of R containing B(Q (R)), are also quasi-Baer and right FI-extending; (ii) lying over, going up, and incomparability of prime ideals, various regularity conditions, and classical Krull dimension transfer between R and RB(Q (R)); and (iii) the existence of a boundedly centrally closed hull for every C * -algebra and a complete characterization for an intermediate C * -algebra between a C * -algebra A and its local multiplier C * -algebra M loc (A) to be boundedly centrally closed. some types of information between R and itself. The disparity between R and Q (R) is evident from the following examples. First take R = Z Q 0 Z , where Z and Q denote the ring of integers and the field of rational numbers, respectively. The ring R is neither right nor left Noetherian and its prime radical is nonzero. However, Q (R) is simple and Artinian. Next, take R to be a domain which does not satisfy the right Ore condition. Then Q (R) is a simple right self-injective (von Neumann) regular ring which has an infinite set of orthogonal idempotents and an unbounded nilpotent index. The vast disparity between R and Q (R) in these examples limits the transfer of information between them.This disparity has motivated us to consider rings from a "distinguished" class that are intermediate between the base ring R and Q (R) or E(R R ). Such an intermediate ring T from a distinguished class possesses the (desirable) properties that identify the class, and since it is "intermediate" it is generally closer to R than either Q (R) or E(R R ). Hence there is some hope that the desirable properties of the specific class and the closeness of T to R will enable a significant transfer of information. Usually this information transfer can be enhanced by: (1) choosing a distinguished class that generalizes some property (or properties) or is related to the class of right self-injective rings;(2) finding (if it exists) a "minimal" element (right ring hull) from the distinguished class; or (3) finding (if they exist) elements of the distinguished class that are "minimally" generated by R and some subset of E(R R ) (pseudo right ring hull).In this paper, using the general approach and the theory of ring hulls introduced in [19], we focus on ring hulls belonging to the class of quasi-Baer rings or the class of right FI-e...

show abstract

“…The (quasi-)Baer conditions are left -right symmetric. It is well known that R is a quasi-Baer if and only if M n (R) is quasi-Baer if and only if T n (R) is a quasi-Baer ring (see [2], [7], [8], [13] and [18]). An idempotent e of a ring R is called left (resp., right) semicentral if ae = eae (resp., ea = eae) for all a ∈ R. It can be easily checked that an idempotent e of R is left (resp., right) semicentral if and only if eR (resp., Re) is an ideal.…”

Section: Introductionmentioning

confidence: 99%

Annihilator Conditions related to the quasi-Baer Condition

Taherifar¹

2016

HJMS

View full text Add to dashboard Cite

We call a ring R an EGE -ring if for each I ¢ R, which is generated by a subset of right semicentral idempotents there exists an idempotent e such that r(I) = eR. The class EGE includes quasi-Baer, semiperfect rings (hence all local rings) and rings with a complete set of orthogonal primitive idempotents (hence all Noetherian rings) and is closed under direct product, full and upper triangular matrix rings, polynomial extensions (including formal power series, Laurent polynomials, and Laurent series) and is Morita invariant. Also we call R an AEring if for each I ¢ R, there exists a subset S ⊆ Sr(R) such that r(I) = r(RSR). The class AE includes the principally quasi-Baer ring and is closed under direct products, full and upper triangular matrix rings and is Morita invariant. For a semiprime ring R, it is shown that R is an EGE (resp., AE)-ring if and only if the closure of any union of clopen subsets of Spec(R) is open (resp., Spec(R) is an EZ-space).

show abstract

Quasi-Baer ring extensions and biregrular rings

Cited by 37 publications

References 27 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*

Hulls of semiprime rings with applications to C∗-algebras

Annihilator Conditions related to the quasi-Baer Condition

Contact Info

Product

Resources

About