Some Pathology for Radicals in Non-Associative Near-Rings

It is well-known that every semisimple class in the class of all associative rings or alternative rings is hereditary, this being an easy consequence of the Anderson-Divinsky-Suliński property (cf. [1]). Dropping the associativity, one get degenerate radicals in the sense that a radical class has a hereditary semisimple class if and only if it is an A-radical, i.e. the radical only depends on the structure of the underlying abelian groups (cf. [4]a and [5]). In the class of all near-rings (or abelian near-rings) the situation is not as bad, here we have both hereditary and non-hereditary semisimple classes, see, for example [3]. Dropping the associativity in this case, the result is even worse than in that of non-associative rings. We show that in the class of all abelian, not-necessatily associative zero-symmetric near-rings the only radicals with hereditary semisimple classes are the two trivial radical classes.