Jordan type of a k[C(p)xC(p)]-module

Let E be the elementary abelian group C(p)xC(p), k a field of characteristic p, M a finite dimensional module over the group algebra k[E] and J the Jacobson radical J of k[E]. We prove that the decomposition of M when considered as a k[< 1+ x >]-module for a p-point x in J is well defined modulo J(p).