IMI/Publicaţii/BASM/Ediţii/BASM n.2 (8), 1992/

On the extension of a pseudonorm of a ring to its field of quotients. (Russian)

Authors: Arnautov Vladimir


Let $R$ be an associative ring, $S$ is such an element from $R$ that $rs\ne 0$ and $sr\ne 0$ for any $0\ne r\in R$ and $s^kR\subseteq Rs$ for some natural number $k$; $S=\{s^i | i=1,2,\dots \}$. Pseudonorm $\zeta$ of ring $R$ we may extend to pseudonorm on rings quotients $R_s$ of ring $R$ on multiplicative system $S$ if and only if
\inf \left \{ \frac{\zeta (rs)}{\zeta(r)}, \frac{\zeta (sr)}{\zeta (r)}
\bigm | 0\ne r\in R \right \} >0.
The given example shows that the demand that $s^k R\subseteq Rs$ for some natural number $k$ can't be substituted for demand that in ring $R$ the right condition Ore is fulfilled concerning multiplicative system $S=\{s^i|i=1,2,\dots \}$.