Let R be a commutative ring, I(R) be the set of all ideals of R and S be a subset of I^*(R) = I(R) -{0} We define a Cayley sum digraph of ideals of R, denoted by Cay^+(I(R),S) as a directed graph whose vertex set is the set I(R) and, for every two distinct vertices I and J, there is an arc from I to J, whenever I + K = J, for some ideal K in S . Also, the Cayley sum graph Cay^+(I(R); S ) is an undirected graph whose vertex set is the set I(R) and two distinct vertices I and J are adjacent whenever I + K = J or J + K = I, for some ideal K in S . In this paper, we study some basic properties of the graphs Cay^-(I(R); S ) and Cay^+(I(R); S ) such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of Cay^+(I(R); S ) and also we provide some characterization for rings R whose Cayley sum graphs have genus one.

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