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I wonder whether someone knows if the Witt cancellation theorem also holds for the rings $\mathbb{Z}/{p^e \mathbb{Z}}$ where $p$ is an odd prime and $e \in \mathbb{N}$, i.e. for example, let $G = diag(p^{e-1}, x_1, ..., x_{n-1})$ and $H = diag(p^{e-1}, y_1, ..., y_{n-1})$ be matrices over $\mathbb{Z}/{p^e \mathbb{Z}}$, such that there exists a matrix $V \in \mathbb{Z}^{n \times n}$ with $det(V)$ coprime to $p$ (i.e. $V$ is invertible over $\mathbb{Z}/{p^e \mathbb{Z}}$) such that $V^T G V \equiv H \mod p^e$ is it also true that one can then "cancel" the first $p^{e-1}$-entry, i.e. is it correct that there exists a matrix $W \in \mathbb{Z}^{n-1 \times n-1}$ with $det(W)$ coprime to $p$ such that $W^T X W \equiv Y \mod p^e$ where $X = diag(x_1, ..., x_{n-1})$ and $Y = diag(y_1, ..., y_{n-1})$?

I already found a paper that claims that it would be possible if the two forms were unimodular (i.e. if $G, H$ were invertible over $\mathbb{Z}/{p^e \mathbb{Z}}$) but in the above situation they are not. Also, some computer experiments with small matrices seem to indicate that it might work :D

Thanks in advance, cheers.

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