## 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 ${W_A} \approx {D_A}$.## References

- Jan-Erik Björk,
*Rings satisfying certain chain conditions*, J. Reine Angew. Math.**245**(1970), 63–73. MR**277562**, DOI 10.1515/crll.1970.245.63 - Paul Moritz Cohn,
*Skew field constructions*, London Mathematical Society Lecture Note Series, No. 27, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR**0463237** - John H. Cozzens,
*Homological properties of the ring of differential polynomials*, Bull. Amer. Math. Soc.**76**(1970), 75–79. MR**258886**, DOI 10.1090/S0002-9904-1970-12370-9 - Carl Faith,
*Algebra. II*, Grundlehren der Mathematischen Wissenschaften, No. 191, Springer-Verlag, Berlin-New York, 1976. Ring theory. MR**0427349** - Stephen M. Ginn,
*A counter-example to a theorem of Kurshan*, J. Algebra**40**(1976), no. 1, 105–106. MR**412229**, DOI 10.1016/0021-8693(76)90090-9 - S. M. Ginn and P. B. Moss,
*Finitely embedded modules over Noetherian rings*, Bull. Amer. Math. Soc.**81**(1975), 709–710. MR**369424**, DOI 10.1090/S0002-9904-1975-13831-6 - Baxter Johns,
*Annihilator conditions in Noetherian rings*, J. Algebra**49**(1977), no. 1, 222–224. MR**453808**, DOI 10.1016/0021-8693(77)90282-4 - R. P. Kurshan,
*Rings whose cyclic modules have finitely generated socle*, J. Algebra**15**(1970), 376–386. MR**260780**, DOI 10.1016/0021-8693(70)90066-9 - B. L. Osofsky,
*On twisted polynomial rings*, J. Algebra**18**(1971), 597–607. MR**280521**, DOI 10.1016/0021-8693(71)90142-6 - Richard Resco,
*Division rings and $V$-domains*, Proc. Amer. Math. Soc.**99**(1987), no. 3, 427–431. MR**875375**, DOI 10.1090/S0002-9939-1987-0875375-3

## 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