Simple and semisimple near-rings

Algebraic systems with addition and multiplication which satisfy all of the skew-field axioms except possibly one of the distributive laws have been studied occasionally. Dickson 1 gave examples which showed that such systems-called Fastkorper by Zassenhauscan actually be nondistributive. Zassenhaus 2 proved that Dickson's examples include all but seven of the finite Fastkorper which are not skew-fields. All continuous Fastkorper of finite degree over the reals were found by Kalscheuer 3. Reidemeister 4 connected Fastkorper with the geometry of webs. All these results are, in a certain sense, extensions of well known facts about fields. This paper makes an analogous extension of part of the theory of semisimple rings to semisimple Fastringe or near-rings. DEFINITION. A set N of elements which can be added and multiplied is said to form a near-ring if 1. the elements form a group under addition, 2. the multiplication of elements is associative, 3. nl(n2+n3) nln2+nln3 for any selection of elements ni, n2, and n3 from N. It follows from this definition that a near-ring satisfies all of the usual ring axioms with the possible exceptions of the right distributive law and the commutative law of addition. The most natural example of near-rings is given by the mappings of a group (written additively) into itself. If the mappings are added by adding images and multiplied by iteration they form a near-ring.' A near-ring homomorphism is a mapping r of a near-ring N into a near-ring N' such that