On feckly clean rings

A ring $R$ is feckly clean provided that for any $a\in R$ there exists an element $e\in R$ and a full element $u\in R$ such that $a=e+u, eR(1-e)\subseteq J®$\mathrm{.}Weprovethataring$R$isfecklycleanifandonlyifforany$a\in R$,thereexistsanelement$e\in R$suchthat$V(a)\subseteq V(e), V(1-a)\subseteq V(1-e)$and$eR(1-e)\subseteq J®$, if and only if for any distinct maximal ideals $M$ and $N$, there exists an element $e\in R$ such that $e\in M,1-e\in N$ and $eR(1-e)\subseteq J(R)$, 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.