Derivable maps of prime rings

الملخص EN

Our active aim in this paper is to prove the following Let Ŕ be a ring having an idempotent element e(e 0,e 1) .

Suppose that R is a subring of Ŕ which satisfies : (i) eR R and Re 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 (R ) R ; i, j 1, 2.

ij ij Then D is additive.

This extend Daif's result to the case R need not contain any non-zero idempotent element.