On near-ring idempotents and polynomials on direct products of Omega-groups

Let N be a zero-symmetric near-ring with identity, and let Gamma be a faith ful tame N-group. We characterize those ideals of Gamma that are the range of some idempotent element of N. Using these idempotents, we show that the polynomials on the direct product of the finite Omega -groups V-1, V-2,..., V-n can be studied componentwise if and only if Pi (n)(i=1) V-i has no ske w congruences.