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 , 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 Mloc(A) to be boundedly centrally closed.