Additive maps on rings behaving like derivations at idempotent-product elements

For every ring R with the unit 1 containing a nontrivial idempotent P. we describe the additive maps 8 from R into itself which behave like derivations, and show that derivations on such kinds of rings can be determined by the action on the elements A. B is an element of R with AB = 0, AB = P and AB = 1 respectively. Those results of An and Hou [R. An, J. Hou, Characterizations of derivations on triangular rings: additive maps derivable at idempotents, Linear Algebra Appl. 431 (2009) 1070-1080], Bresar [M. Bresar, Characterizing homomorphisms, multipliers and derivations in rings with idempotents, Proc. Roy. Soc. Edinburgh. Sect. A. 137 (2007) 9-21] and Chebotar et al. [M.A. Chebotar, W.-F. Ke, P.-H. Lee, Maps characterized by action on zero products, Pacific J. Math. 216 (2) 2004 217-228] are improved. (C) 2010 Elsevier B.V. All rights reserved.