A semi-isometric isomorphism on a ring of matrices

Let $(R,\xi)$ be a pseudonormed ring and $R_n$ be a ring of matrices over the ring $R$. We prove that if $1\leq\gamma,\sigma\leq\infty$ and $\frac 1\gamma+\frac1\sigma\geq1$, then the function $\eta_{\xi,\gamma,\sigma}$ is a pseudonorm on the ring $R_n$. Let now $\varphi\colon(R,\xi)\to(\overline R,\overline\xi)$ be a semi-isometric isomorphism of pseudonormed rings. We prove that $\Phi\colon(R_n,\eta_{\xi,\gamma,\sigma})\to(\overline R_n,\eta_{\overline\xi,\gamma,\sigma})$ is a semi-isometric isomorphism too for all $1\leq\gamma,\sigma\leq\infty$ such that $\frac1\gamma+\frac1\sigma\ge1$.