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 satisfy 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 such that a(m)(1 - e) is an element of a(m)N(r)(R). For example, if R is right weakly x-regular or every idempotent of R is central, then R satisfies WCl. Many authors have considered the equivalence of condition pm (i.e., every prime ideal is maximal) with various generalizations of von Neumann regularity over certain classes of rings including commutative, PI, right due, and reduced. In the context of weakly pi-regular 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 conditions are equivalent: (1) R is right weakly pi-regular; (2) R/N-r(R) is right weakly pi-regular and R satisfies WCI; (3) R/N-r(R) is biregular and 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. Let R be a ring such that N-r(R) is completely semiprime and R satisfies WCI. Then the following conditions are equivalent: (1) R is right weakly 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 ideal of R such that N-r(R) subset of or equal to P, then P = <(O)over bar p>.