Hulls of Ring Extensions

Birkenmeier, Gary F.; Park, Jae Keol; Rizvi, S. Tariq

doi:10.4153/cmb-2010-065-9

Cited by 5 publications

(3 citation statements)

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“…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: 98%

“…We note that the quasi-Baer and FI-extending hulls of a semiprime ring are well behaved with respect to various ring extensions (e.g., group rings, matrix rings, etc.) [21].…”

mentioning

confidence: 95%

“…We note that the quasi-Baer and FI-extending hulls of a semiprime ring are well behaved with respect to various ring extensions (e.g., group rings, matrix rings, etc.) [21].The final section of the paper is devoted to applications of our results to a ring with involutionin particular to a C * -algebra. The study of the local multiplier algebra, M loc (A), for a C * -algebra A was initiated by Elliott in [31] and Pedersen in [52] to study * -automorphisms and derivations.…”

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confidence: 97%

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Hulls of semiprime rings with applications to C∗-algebras

2009

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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...

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Section: Introductionmentioning

confidence: 98%

“…We note that the quasi-Baer and FI-extending hulls of a semiprime ring are well behaved with respect to various ring extensions (e.g., group rings, matrix rings, etc.) [21].…”

mentioning

confidence: 95%

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confidence: 97%

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