Let R be a commutative local ring which is not necessarily noetherian. Denote by J ( R ) the Jacobson radical of R and Z ( R ) the zero-divisor elements of R . A local R is called Z -local if J ( R ) = Z ( R ) and J ( R )2 = 0. In this paper the structure of a For a commutative ring R, assume that c is a nonzero element of Z(R) with the property that Cz(R ) = 0 . A local ring R is called c-local if Z(R)2 = {0, c},Z(R )3 = 0, and Xz(R) = 0 implies x ? {0, c}. For any finite c-local ring R,it is proved that the ideal m has a minimal generating set which has a c-partition. The structure and with order greater than 2^5 are determined.