Let R be a commutative ring. The regular digraph of ideals of R, denoted by \gama(R), is a digraph whose vertex-set is the set of all non-trivial ideals of R and, for every two distinct vertices I and J, there is an arc from I to J, whenever I contains a non-zero divisor on J. In this paper, we investigate the planarity of \gama(R). We also completely characterize the rings R such that \gama(R) is a ring graph, and the situations under which the genus of \gama(R) is finite. Moreover, we study the independence number and the girth of \gama(R), and also we find all cases that \gama(R) is bipartite.

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