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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A counter example to a conjecture of Johns
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by Carl Faith and Pere Menal PDF
Proc. Amer. Math. Soc. 116 (1992), 21-26 Request permission

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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Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 116 (1992), 21-26
  • MSC: Primary 16P40; Secondary 16P50
  • DOI: https://doi.org/10.1090/S0002-9939-1992-1100651-0
  • MathSciNet review: 1100651