Vol. 49, No. 2, 1973

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Rings of quotients of rings without nilpotent elements

Stuart A. Steinberg

Vol. 49 (1973), No. 2, 493–506

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.

Mathematical Subject Classification
Primary: 16A08
Received: 14 August 1972
Revised: 1 February 1973
Published: 1 December 1973
Stuart A. Steinberg