Some results in prime rings involving endomorphisms

Abstract In this paper, we generalize some results from [M. Ashraf and S. Ali, On left multipliers and the commutativity of prime rings, Demonstr. Math. 41 2008, 4, 763–771] and present our contribution to the study of endomorphisms. In particular, we prove that a prime ring must be an integral domain if it admits endomorphisms and multipliers satisfying some algebraic identities on non-zero ideals. As a consequence of our main theorems, many known results can either be generalized or deduced. Furthermore, an example is given to show that the necessity of the primeness assumption imposed on the hypotheses of various theorems cannot be unnecessary.