On the Ideals and Automorphisms of Non-associative Rings

Introduction. Relations between the multiplication ring of a ring (= non-associative ring)1 and the ring itself have been pointed out by a number of writers [l-7]. In the present note it is shown that the ideal lattice of a ring with unit element is isomorphic to the sublattice of all right ideals of the multiplication ring which contain the annihilator of the unit. A corresponding result is obtained for the right ideal lattice. These results generalize some conditions of R. D. Schäfer [7] which describe the simplicity or right simplicity of an algebra with unit element in terms of the right ideal structure of the multiplication ring. We do not quite assume the existence of a unit. (See the paragraph preceding Lemma 2.) By adjoining a unit element, we are able to derive similar, but not quite so precise, results for arbitrary rings. We also give considerably simplified proofs of the results2 of Schäfer [7] which concern the automorphisms of rings with unit. Here our simplification consists in avoiding the so-called "reconstruction" of a ring with unit from its multiplication ring. Again we do not quite require a unit.