RADICAL THEORY IN VARIETIES OF NEAR-RINGS IN WHICH THE CONSTANTS FORMAN IDEAL

Not all the good properties of the Kurosh-Amitsur radical theory in th e variety of associative rings are preserved in the bigger variety of near-rings. In the smaller and better behaved variety of 0-symmetric n ear-rings the theory is much more satisfactory. In this note we show t hat many of the results of the 0-symmetric near-ring case can be exten ded to a much bigger variety of near-rings which, amongst others, incl udes all the 0-symmetric as well as the constant near-rings. The varie ties we shall consider are varieties of near-rings, called Fuchs varie ties, in which the constants form an ideal. The good arithmetic of suc h varieties makes it possible to derive more explicit conditions (i) f or the subvariety of constant near-rings to be a semisimple class (or equivalently, to have attainable identities), (ii) for semisimple clas ses to be hereditary. We shall prove that the subvariety of 0-symmetri c 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.