Anderson and Smith [1] defined weakly prime ideal of a commutative ring with identity and considered the necessary and sufficient condition for a ring in which every proper ideal is weakly prime. Hirano et al. [2] generalized the definition of weakly prime ideal for any ring (not necessary commutative nor with identity) and proved some properties of ring in which every proper ideal is weakly prime (then it is called fully weakly prime ring). Necessary and sufficient conditions for fully weakly prime ring are also given. Hirano et al. [2] also studied some possibilities of nilpotent radical of fully weakly prime ring. In this paper, we establish a necessary and sufficient condition for a fully weakly prime ring using the characteristics of its nilpotent radicals.

fully weakly prime ring, weakly prime ideal, nilpotent radical.