Derivable Maps of Prime Rings

Our active aim in this paper is to prove the following Let Ŕ be a ring having an idempotent element e(e neq 0,e neq 1) . Suppose that R is a subring of Ŕ which satisfies: (i) eR subseteq R and Re subseteq R . (ii) xR = 0 implies x = 0 . (iii ) eRx = 0 implies x = 0( and hence Rx = 0 implies x = 0) . (iv) exeR(1- e)  0 implies exe = 0 . If D is a derivable map of R satisfying D(Rij ) subseteq Rij =1,2. Then D is additive. This extend Daif s result to the case R need not contain any non-zero idempotent element.