Goldie’s theorem for alternative rings

Abstract

It is known that the socle of a semiprime Goldie ring is generated by a central idempotent and that a prime Goldie ring with a nonzero socle is a simple artinian ring. We prove the extension of these results to alternative rings. We also give an analogue of Goldie's theorem for alternative rings. A Goldielike theorem was obtained earlier by the authors for noetherian alternative rings by a quite different method.

Literature Survey on Non-Associative Rings and Developments

“…In 1994, Essannounia and Kaidi [48] discovered that the socle of a semiprime Goldie ring is generated by a central idempotent and that a prime Goldie ring with a nonzero socle is a simple artinian ring. They also extended these results to alternative rings.…”

Algebraic lattices and nonassociative structures

“…H. Essannouni and A. Kaidi gave in [3] a Goldie-like Theorem for alternative rings (see also [2] for a former version of this theorem by the same authors). We now provide a new proof of this result based on Slater's theorem for prime nondegenerate alternative algebras and on the fact that a semiprime alternative algebra not containing infinite direct sums of right ideals satisfies the chain conditions on the annihilators of its ideals, and hence it is an essential subdirect sum of a finite number of prime nondegenerate alternative algebras each of which does not contain infinite direct sums of right ideals.…”