Regularity conditions and the simplicity of prime factor rings Original Research Article

R denotes a ring with unity and Nr(R) its nil radical. R is said to satisfy conditions: 1. (1) pm(Nr) if every prime ideal containing Nr(R) is maximal; 2. (2) WCI if whenever a,e ε R such that e = e2, eR + Nr(R) = RaR + Nr(R), and xe − ex ε Nr(R) for any x ε R, then there exists a positive integer m such that am(1 − e) ε amNr(R). For example, if R is right weakly π-regular or every idempotent of R is central, then R satisfies WCI. 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 duo, and reduced. In the context of weakly π-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 Nr(R) completely semiprime. Then the following conditions are equivalent: (1) R is right weakly π-regular; (2) R/Nr(R) is right weakly π-regular and R satisfies WCI; (3) R/Nr(R) is biregular and R satisfies WCI; (4) for each χ ε R there exists a positive integer m such that R = RχmR + r(χm). Theorem II. Let R be a ring such that Nr(R) is completely semiprime and R satisfies WCI. Then the following conditions are equivalent: (1) R is right weakly π-regular; (2) R/Nr(R) is right weakly π-regular; (3) R/Nr(R) is biregular; (4) R satisfies pm(Nr); (5) if P is a prime ideal such that Nr(R/P) = 0, then R/P is a simple domain; (6) for each prime ideal of R such that Nr(R) subset of or equal to P, then image.