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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A counter example to a conjecture of Johns


Authors: Carl Faith and Pere Menal
Journal: Proc. Amer. Math. Soc. 116 (1992), 21-26
MSC: Primary 16P40; Secondary 16P50
MathSciNet review: 1100651
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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}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1100651-0
PII: S 0002-9939(1992)1100651-0
Article copyright: © Copyright 1992 American Mathematical Society