A counter example to a conjecture of Johns

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 ${W_A}$ is injective and $A$ embeds in the endomorphism ring $\operatorname {End} ({W_A})$. Then the counter example is the trivial extension $R = A \ltimes W$ 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 $A = D{ \otimes _k}k(x)$, where $k(x)$ is the field of rational functions, has the desired properties, with ${W_A} \approx {D_A}$.