Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1992-1100651-0
MathSciNet review: 1100651
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 ${W_A} \approx {D_A}$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 16P40, 16P50

Retrieve articles in all journals with MSC: 16P40, 16P50


Additional Information

Article copyright: © Copyright 1992 American Mathematical Society