On feckly clean rings

A ring R is feckly clean provided that for any a R there exists an element e R and a full element u R such that a = e + u, eR(1 - e) J(R). We prove that a ring R is feckly clean if and only if for any a R, there exists an element e R such that V(a) V(e), V(1 - a) V(1 - e) and eR(1 - e) J(R), if and only if for any distinct maximal ideals M and N, there exists an element e R such that e M, 1 - e N and eR(1 - e) J(R), if and only if J-spec(R) is strongly zero-dimensional, if and only if Max(R) is strongly zero-dimensional and every prime ideal containing J(R) is contained in a unique maximal ideal. More explicit characterizations are also discussed for commutative feckly clean rings. © 2015 World Scientific Publishing Company.