Noetherian ring extensions with trace conditions

Warfield, Robert B.

doi:10.1090/S0002-9947-1992-1080737-4

Noetherian ring extensions with trace conditions
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Trans. Amer. Math. Soc. 331 (1992), 449-463 Request permission

Abstract:

Finite ring extensions of Noetherian rings with certain restrictions on the corresponding trace ideals are studied. This setting includes finite free extensions and extensions arising from actions of finite groups when the order of the group is invertible. In this setting we establish the following results which were previously obtained (for finite extensions without trace conditions) only under strong restrictions on the rings involved. Let $R \subset S$ be an extension of Noetherian rings such that $S$ is finitely generated as a left $R$-module and such that the left trace ideal of $S$ in $R$ is equal to $R$. If $S$ is right fully bounded, or is a Jacobson ring, then $R$ has the same property; furthermore, $R$ and $S$ have the same classical Krull dimension. If $S$ is finitely generated as both a right and a left $R$-module, if both trace ideals of $S$ in $R$ are equal to $R$, and if $S$ satisfies the strong second layer condition, then this condition also holds in $R$. Finally, we compare the link graphs of $R$ and $S$

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