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

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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 (not a field) with a unique simple right module such that is injective and embeds in the endomorphism ring . Then the counter example is the trivial extension of and . The ring exists by a theorem of Resco using a theorem of Cohn. Specifically, if is any countable existentially closed field with center , then the right and left principal ideal domain defined by , where is the field of rational functions, has the desired properties, with .

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1100651-0

Article copyright:
© Copyright 1992
American Mathematical Society