In this paper, we study the algebraic structure of Z(p)(r) Z(p)(s)-additive codes which are Z-submodules where p is prime, and r and s are positive integers. Z(p)(r) Z(p)(s) additive codes naturally generalize Z(2)Z(4) and Z(2)Z(2)s-additive codes which have been introduced recently. The results obtained in this work generalize a great amount of the studies done on additive codes. Especially, we determine the standard forms of generator and parity-check matrices for this family of codes. This leads to the identification of the type and the cardinalities of these codes. Furthermore, we present some bounds on the minimum distance and examples that attain these bounds. Finally, we present some illustrative examples for some special cases.