Throughout, R is an associative ring with identity, and C(R), J(R), U(R) and Nil(R) denote The center, The Jacobson radical, The unit group and The set of all nilpotent elements of R, respectively. An element a of a ring R is called nil-clean if a = e amp;#??;b where e? = e ? R and b is a nilpotent element of R; if furTher eb = be, Then a is called strongly nil-clean. The ring R is called nil-clean (resp., strongly nil-clean) if each of its elements is nil-clean (resp., strongly nil-clean). several equivalent conditions of a strongly nil-clean element is given: an element a ? R is strongly nil-clean if and only if a is strongly ??regular and a ? a^? is nilpotent if and only if a is uniquely strongly clean and a ? a? is nilpotent. This follows that a ring R is strongly nil-clean if and only if R/J(R) is a boolean ring and J(R) is a nil ideal.
