Definition 29.47.1. Let $A$ be a ring.

We say $A$ is

*seminormal*if for all $x, y \in A$ with $x^3 = y^2$ there is a unique $a \in A$ with $x = a^2$ and $y = a^3$.We say $A$ is

*absolutely weakly normal*if (a) $A$ is seminormal and (b) for any prime number $p$ and $x, y \in A$ with $p^ px = y^ p$ there is a unique $a \in A$ with $x = a^ p$ and $y = pa$.

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