The structure of Johns rings

Faith, Carl; Menal, Pere

doi:10.1090/S0002-9939-1994-1231294-8

The structure of Johns rings
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by Carl Faith and Pere Menal

Proc. Amer. Math. Soc. 120 (1994), 1071-1081

DOI: https://doi.org/10.1090/S0002-9939-1994-1231294-8

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Erratum: Proc. Amer. Math. Soc. 125 (1997), 1247-1247.

Abstract:

In this paper we continue our study of right Johns rings, that is, right Noetherian rings in which every right ideal is an annihilator. Specifically we study strongly right Johns rings, or rings such that every $n \times n$ matrix ring ${R_n}$ is right Johns. The main theorem (Theorem 1.1) characterizes them as the left ${\text {FP}}$-injective right Noetherian rings, a result that shows that not all Johns rings are strong. (This first was observed by Rutter for Artinian Johns rings; see Theorem 1.2.) Another characterization is that all finitely generated right $R$-modules are Noetherian and torsionless, that is, embedded in a product of copies of $R$. A corollary to this is that a strongly right Johns ring $R$ is preserved by any group ring $RG$ of a finite group (Theorem 2.1). A strongly right Johns ring is right $FPF$ (Theorem 4.2).

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