Radicals and Idempotents IV: Alternative Rings

This paper is a sequel to a series in which the authors have explored the behaviour of radicals of associative rings with respect to subrings eAe, where e is an idempotent in a ring A. A 1974 paper of Anderson contains the result that if a radical class of associative rings is such that the radical of every ring contains all radical one-sided ideals, then for a semi-simple ring A, each eAe is also semi-simple. Anderson asks whether this result is also true for alternative rings. By a close examination of some properties of a split Cayley-Dickson algebra, it is shown here that the result is not true for alternative rings. It is further shown that some other properties of associative rings do not extend to the alternative case, but many do so. Notably the definition and properties of q-central idempotents are similar in the alternative case to those for associative rings.