Semiprime alternative rings with A.C.C.

Literature Survey on Non-Associative Rings and Developments

“…However, this was not so for prime right alternative rings in general. In 1988, Essannouni and Kaidi [47] proved the natural extension to alternative rings of the classical Goldie theorem for semiprime associative 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.…”

A Result on Assosymmetric Rings Satisfying the Ascending Chain Condition

“…Let R be as in the main theorem and let Q be its right quotient ring relative to S. Let x be a regular element of R. It is easy to see that x is a right regular in Q. By the structure theorem of semisimple artinian alternative rings [5,7], it is easy to see that every right regular element of Q is invertible which is also applicable to the assosymmetric ring. Thus x is invertible in Q.…”