Assume that $R$ is a discrete valuation ring with valuation $v:R\rightarrow \mathbb{N}.$ Let $K$ be the field of fractions of $R$.
Is there a meaningful way to extend the valuation to the matrices $K^{n\times n}$ over $K$?
By “meaningful” I mean a map $v:K^{n\times n}\rightarrow \mathbb{Z}^{n\times n}$ such that a) $v(MN)=v(M)+v(N)$ and b) for any diagonal matrix we have $v(\mathrm{diag}(x_1,\dots,x_n))=\mathrm{diag}(v(x_1),\dots,v(x_n))$.