A Counter Example to a Conjecture of Johns

Abstract

Abstract.In this paper, we construct a counter example to a conjecture of Johns to the effect that a right Noetherian ring in which every right ideal is an annihilator is right Artinian. Our example requires the existence of a right Noetherian domain A (not a field) with a unique simple right module W such that WA is injective and A embeds in the endomorphism ring End(WA). Then the counter example is the trivial extension R = A x W of A and W . The ring A exists by a theorem of Resco using a theorem of Cohn. Specifically, if D is any countable existentially closed field with center k , then the right and left principal ideal domain defined by A = D®k k(x), where k(x) is the field of rational functions, has the desired properties, with WA « DA .