Radical Theory in Varieties of Near-Rings in which the Constants form an Ideal

Not all the good properties of the Kurosh-Amitsur radical theory in the variety of associative rings are preserved in the bigger variety of near-rings. In the smaller and better behaved variety of 0-symmetric near-rings the theory is much more satisfactory. In this note we show that many of the results of the 0-symmetric near-ring case can be extended to a much bigger variety of near-rings which, amongst others, includes all the 0-symmetric as well as the constant near-rings. The varieties we shall consider are varieties of near-rings, called Fuchs varieties, in which the constants form an ideal. The good arithmetic of such varieties makes it possible to derive more explicit conditions (i) for the subvariety of constant near-rings to be a semisimple class (or equivalently, to have attainable identities), (ii) for semisimple classes to be hereditary. We shall prove that the subvariety of 0-symmetric near-rings has attainable identities in a Fuchs variety, and extend the theory of overnilpotent radicals of 0-symmetric near-rings to the largest Fuchs variety F. The near-ring construction of [7] will play a decisive role in our investigations.