In this note we generalize some results from finite fields to Galois rings which are finite extensions of the ring Zpm of integers modulo pm where p is a prime and m ≥ 1.

We consider a commutative ring $R$ with identity and a positive integer $N$. We characterize all the 3-tuples $(L_1,L_2,L_3)$ of linear transforms over $R^N$, having the “circular convolution” property, i.eṡuch that $x*y=L_3(L_1\left(x\right)\otimes L_2\left(y\right))$ for all $x,y\in R^N$.

Suppose that $R\subset S$ are regular local rings which are essentially of finite type over a field $k$ of characteristic zero. If $V$ is a valuation ring of the quotient field $K$ of $S$ which dominates $S$, then we show that there are sequences of monoidal transforms (blow ups of regular primes) $R\to R_1$ and $S\to S_1$ along $V$ such that $R_1\to S_1$ is a monomial mapping. It follows that a morphism of nonsingular varieties can be made to be a monomial mapping along a valuation, after blow ups of nonsingular subvarieties.