  This paper investigates the application of the theoretical algebraic notion
of a separable ring extension, in the realm of cyclic convolutional codes or,
more generally, ideal codes. We work under very mild conditions, that cover all
previously known as well as new non trivial examples. It is proved that ideal
codes are direct summands as left ideals of the underlying non-commutative
algebra, in analogy with cyclic block codes. This implies, in particular, that
they are generated by an idempotent element. Hence, by using a suitable
separability element, we design an efficient algorithm for computing one of
such idempotents.
