Finite extensions of rings

Cortzen, Barbara; Small, Lance W.

doi:10.1090/S0002-9939-1988-0954983-6

Finite extensions of rings
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by Barbara Cortzen and Lance W. Small

Proc. Amer. Math. Soc. 103 (1988), 1058-1062

DOI: https://doi.org/10.1090/S0002-9939-1988-0954983-6

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Abstract:

The paper concerns some cases of ring extensions $R \subset S$, where $S$ is finitely generated as a right $R$-module and $R$ is right Noetherian. In ${\text {\S }}1$ it is shown that if $R$ is a Jacobson ring, then so is $S$, with the converse true in the ${\text {PI}}$ case. In ${\text {\S }}2$ we show that if $S$ is semiprime ${\text {PI}}$, $R$ must also be left (as well as right) Noetherian and $S$ is finitely generated as a left .$R$-module. ${\text {\S }}3$ contains a result on $E$-rings.

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