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

Let $N$ be a zero-symmetric near-ring with identity, and let $\Gamma$ be a faithful 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 $V_1, V_2, \ldots, V_n$ can be studied componentwise if and only if $\prod_{i=1}^n V_i$ has no skew congruences.