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For a finite ring R with identity and a finite unital Ä-module V we call C(R) = {/: K-> V\f(av) = af(v) for all a 6 R, v e V) the nearring centralizer of R. We investigate the structure of C(R) and obtain a characterization of those rings R for which C(R) is a simple nonring.

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Ž. Let N be a right near-ring with identity such that N, q is abelian. Because N enjoys the right distributive property, every right multiplication map on N is an Ž . endomorphism of N, q . The set of all right multiplication maps on N generates Ž . a ring ‫,ޒ‬ a subring of the ring End N . The structure of ‫ޒ‬ is investigated when N is a finite simple near-ring and when N is a finite centralizer near-ring. ᮊ 1996Academic Press, Inc.

Abstract. Let J be a noncommutative Jordan algebra with 1. If / has two orthogonal idempotents e and / such that 1 =e+/and such that the Peirce 1-spaces of each are Jordan division rings, then J is said to have capacity two. We prove that a simple noncommutative Jordan algebra of capacity two is either a Jordan matrix algebra, a quasi-associative algebra, or a type of quadratic algebra whose plus algebra is a Jordan algebra determined by a nondegenerate symmetric bilinear form.