The first theorem gives a
number of characterizations of when a ring with zero right singular ideal has a
strongly regular right quotient ring. This result (and also Theorem 2) is a
generalization of a similar theorem of F. W. Anderson for a certain class of
lattice-ordered rings and a theorem of G. Renault for reduced rings (i.e., rings
without nilpotent elements). As a corollary one obtains a characterization of when a
semiprime ring has a strongly regular right quotient ring similar to Utumi’s
characterization of when a ring has a regular right quotient ring. Also, some theorems
on commutative regular rings are extended to strongly regular rings and regular rings
that satisfy a polynomial identity. For instance, a reduced ring is regular if and only
if each of its prime homomorphic images is regular. This theorem has been
obtained independently by Herstein, by Snider, and by Wong. Using rings of
quotients some theorems of R. Wiegand are also generalized. It is shown
that the endomorphism ring S of an ideal I of a strongly regular ring R is
strongly regular, and some characterizations of when R is self-injective are
obtained.
