Alternative Rings Without Nilpotent Elements

Abstract

In this paper we show that any alternative ring without nonzero nilpotent elements is a subdirect sum of alternative rings without zero divisors. Andrunakievic and Rjabuhin proved the corresponding result for associative rings by a complicated$^{1}$ process in 1968. Our result extends Andrunakievic and Rjabuhin’s result to the alternative case, and our argument is nearly as simple as in the associative-commutative case. Since right alternative rings of characteristic not 2 without nilpotent elements are alternative, our results extend to such rings as well.