Degree of the kernel of a module map $R^n\rightarrow R^n$ for an Euclidean domain $R$

Let $R$ be an Euclidean domain with the degree function $d$. Let $A\in R^{n\times n}$ be an $n\times n$-matrix with entries in $R$ such that det$(A)=0$. As a module map $A:R^n\rightarrow R^n$, there always exists a kernel element $v\in R^n$ since det$(A)=0$.

Assuming $d(A_{ij})\leq m$ for all $i,j$, is there an explicit bound $k(m,n)$ such that there exists a kernel element $v\in R^n$ satisfying $d(v_i)\leq k(m,n)$?

Edit : $v$ is assumed to be nonzero.

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