REGULARITY CONDITIONS AND THE SIMPLICITY OF PRIME FACTOR RINGS

R denotes a ring with unity and N-r(R) its nil radical. R is said to s atisfy conditions: (1) pm(N-r) if every prime ideal containing N-r(R) is maximal; (2) WCl if whenever a, e is an element of R such that e = e(2), eR + N-r(R) = RaR + N-r(R), and xe - ex is an element of N-r(R) for any x is an element of R, then there exists a positive integer m s uch that a(m)(1 - e) is an element of a(m)N(r)(R). For example, if R i s right weakly x-regular or every idempotent of R is central, then R s atisfies WCl. Many authors have considered the equivalence of conditio n pm (i.e., every prime ideal is maximal) with various generalizations of von Neumann regularity over certain classes of rings including com mutative, PI, right due, and reduced. In the context of weakly pi-regu lar rings, we prove the following two theorems which unify and extend nontrivially many of the previously known results. Theorem I. Let R be a ring with N-r(R) completely semiprime. Then the following condition s are equivalent: (1) R is right weakly pi-regular; (2) R/N-r(R) is ri ght weakly pi-regular and R satisfies WCI; (3) R/N-r(R) is biregular a nd R satisfies WCI; (4) for each chi is an element of R there exists a positive integer m such that R = R chi(m)R + r(chi(m)). Theorem II. L et R be a ring such that N-r(R) is completely semiprime and R satisfie s WCI. Then the following conditions are equivalent: (1) R is right we akly pi-regular; (2) R/N-r(R) is right weakly pi-regular; (3) R/N-r(R) is biregular; (4) R satisfies pm(N-r); (5) if P is a prime ideal such that N-r(R/P) = 0, then R/P is a simple domain; (6) for each prime id eal of R such that N-r(R) subset of or equal to P, then P = <(O)over b ar p>.