Hulls of semiprime rings with applications to C∗-algebras

Abstract

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...

Hulls of Ring Extensions

“…Take e ∈ B(Q(R)) and let J = R ∩ (1 − e)Q(R). Then r Q(R) ( J) = eQ(R) as in the proof of [18,Theorem 3.3].…”

“…In [16] and [18], the ring hull concept with respect to a class of rings was introduced and developed. Let H K (R) denote a ring hull of R with respect to a class K of rings and X(R) denote a ring extension of R. Then it is natural to ask: How does H K (X(R)) compare with X(H K (R))?…”

“…Boundedly centrally closed algebras have been used to obtain a complete description of all centralizing additive mappings on C * -algebras[1] and for investigating the central Haagerup tensor product of multiplier algebras[2].Let A be a unital C * -algebra. Then it is shown in[18, Lemma 4.9(i)] that Q qB (A) is a * -subalgebra of Q b (A), so Q qB (A) is a * -subalgebra of M loc (A). By [18,Lemma 4.12(i)], A is boundedly centrally closed if and only if A ∈ qB.…”

“…. , f n 1 ∈ B(CFM Γ ( Q qB (R))) are mutually orthogonal by[18, Lemma 4.9] (note that f i ∈ B( Q qB (R)) for all i and these are mutually orthogonal) andµ = θ 1 f 1 1 + • • • + θ n f nThus for each entry of the diagonal of µ (or equivalently, each element of RB(Q(R))), say a, there exist diagonal entries θ i (a) of θ i for i = 1, . .…”

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

“…In [14], the quasinite group actions on a semiprime ring and their applications to C * -algebras have been studied (see also [17,18]). Motivated by investigations in [14], in this paper we investigate the right p.q.-Baer property of fixed rings under finite group actions on a given semiprime ring.…”