Let
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}R$
be a commutative ring,
$I(R)$
be the set of all ideals of
$R$
and
$S$
be a subset of
$I^*(R)=I(R)\setminus \{0\}$
. We define a Cayley sum digraph of ideals of
$R$
, denoted by
$\overrightarrow{\mathrm{Cay}}^+ (I(R),S)$
, 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
$J$
, denoted by
$I\longrightarrow J$
, whenever
$I+K=J$
, for some ideal
$K $
in
$S$
. Also, the Cayley sum graph
$ \mathrm{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
$\overrightarrow{\mathrm{Cay}}^+ (I(R),S)$
and
$ \mathrm{Cay}^+ (I(R), S)$
such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of
$ \mathrm{Cay}^+ (I(R), S)$
and also we provide some characterization for rings
$R$
whose Cayley sum graphs have genus one.