On feckly clean rings

A ring R is feckly clean provided that for any a is an element of R there exists an element e is an element of R and a full element u is an element of R such that a = e + u, eR(1 - e) subset of J(R). We prove that a ring R is feckly clean if and only if for any a is an element of R, there exists an element e is an element of R such that V (a) subset of V (e), V (1 - a) subset of V (1 - e) and eR(1 - e) subset of J(R), if and only if for any distinct maximal ideals M and N, there exists an element e is an element of R such that e is an element of M, 1 - e is an element of N and eR(1 - e) subset of 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.