Semi-simple Extensions of Rings

In this paper we investigate the conditions under which a given ring is a sub-ring of a semi-simple ring. For convenience, we say that a ring A is an extension of a ring B, if B is a sub-ring of A. It is found that the existence of a semi-simple extension is equivalent to the vanishing of the extension radical, a two-sided ideal defined analogously to the ordinary radical. In Theorem II we give an intrinsic characterization of the extension radical, where we find that the latter is determined by the addition in the ring and is independent of the multiplication. This result is summarized in Theorem III. For the convenience of the reader, we reproduce here some of the definitions given in the paper mentioned in footnote 1. The radical of a ring A is obtained as the intersection of the annihilators of all simple A -modules. When the radical consists only of the zero element of A, we say that the ring is semi-simple. The radical as defined here contains the ideal classically known as the radical (the sum of all nilpotent left ideals) and is equal to it if one assumes tht A satisfies the descending chain condition on left ideals. A ring which is semisimple in the present sense has then no nilpotent ideals. In order to define the extension radical, we must first introduce the auxiliary notion of a quasi-simple module. If 9ft is an abelian group, denote by £(9ft) the ring of all endomorphisms of 9ft. We say that 9ft is a quasi-simple group if 9ft is a simple E(9ft)-module. An A -module 2ft is a quasi-simple module if the underlying additive group of 9ft is quasi-simple. We shall find the following two lemmas useful.