Hulls of semiprime rings with applications to C*-algebras

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 vertical bar r epsilon R and e epsilon 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 night 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. (C) 2009 Elsevier Inc. All rights reserved.