  It is known that an ideal of a direct product of commutative unitary rings is
directly decomposable into ideals of the corresponding factors. We show that
this does not hold in general for commutative rings and we find necessary and
sufficient conditions for direct decomposability of ideals. For varieties of
commutative rings we derive a Mal'cev type condition characterizing direct
decomposability of ideals and we determine explicitly all varieties satisfying
this condition.
