A~semi-isometric isomorphism on a~ring of matrices

Svetlana Aleschenko

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2014), pp. 74-84.

Résumé

Let $(R,\xi )$ be a pseudonormed ring and $R_n$ be a ring of matrices over the ring $R$. We prove that if $1\le \gamma ,\sigma \le \mathrm{\infty }$ and $\frac{1}{\gamma }+\frac{1}{\sigma }\ge 1$, then the function ${\eta }_{\xi ,\gamma ,\sigma }$ is a pseudonorm on the ring $R_n$. Let now $\phi :(R,\xi )\to (\bar{R},\bar{\xi })$ be a semi-isometric isomorphism of pseudonormed rings. We prove that $\mathrm{\Phi }:(R_n,{\eta }_{\xi ,\gamma ,\sigma })\to ({\bar{R}}_n,{\eta }_{\bar{\xi },\gamma ,\sigma })$ is a semi-isometric isomorphism too for all $1\le \gamma ,\sigma \le \mathrm{\infty }$ such that $\frac{1}{\gamma }+\frac{1}{\sigma }\ge 1$.

Bibliographie

[1] Aleschenko S. A., Arnautov V. I., “Quotient rings of pseudonormed rings”, Buletinul Academiei de Ştiinte a Republicii Moldova, Matematica, 2006, no. 2(51), 3–16 | MR | Zbl

[2] Aleschenko S. A., Arnautov V. I., “Properties of one-sided ideals of pseudonormed rings when taking the quotient rings”, Buletinul Academiei de Ştiinte a Republicii Moldova, Matematica, 2008, no. 3(58), 3–8 | MR | Zbl

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[4] Aleschenko S. A., “Constructions of pseudonormed rings which keep a semi-isometric isomorphism”, Studia Universitatis Moldaviae, Ştiinţe exacte şi economice, 2013, no. 7(67), 19–27