Geodesic
Properties of one-sided ideals of pseudonormed rings when taking the quotient rings
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2008), pp. 3-8.

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Let $\varphi\colon(R,\xi)\to\bigl(\widehat{R},\widehat{\xi}\bigr)$ be an isomorphism of pseudonormed rings. The inequalities $\dfrac{\xi(a\cdot b)}{\xi(b)}\leq\widehat{\xi}(\varphi(a))\leq\xi(a)$ are fulfilled for any $a,b\in R\setminus\{0\}$ iff there exists a pseudonormed ring $\bigl(\widetilde{R},\widetilde{\xi}\bigr)$ such that $(R,\xi)$ is a left ideal in $\bigl(\widetilde{R},\widetilde{\xi}\bigr)$ and the isomorphism $\varphi$ can be extended up to an isometric homomorphism $\widetilde{\varphi}\colon\bigl(\widetilde{R},\widetilde{\xi}\bigr)\to\bigl(\widehat{R},\widehat{\xi}\bigr)$. 
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S. A. Aleschenko; V. I. Arnautov. Properties of one-sided ideals of pseudonormed rings when taking the quotient rings. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2008), pp. 3-8. http://geodesic.mathdoc.fr/item/BASM_2008_3_a0/

[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