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Given a ring R, we investigate a subfunctor Z* of the identity functor on the category of all right R-modules which is defined by is a small module , for any R-module M. We prove that if the ring R satisfies the descending chain condition for right annihilators and is an Artinian ring for every primitive ideal P then for every right R-module M, where S is the left socle of R. Moreover the ring R is semiprime Artinian if and only if R is right bounded, R satisfies the descending chain condition for right annihilators and for some faithful right R-module M.