RO EN
On the extension of a pseudonorm of a ring to its field of quotients. (Russian)
Authors: Arnautov Vladimir
$$
\inf \left \{ \frac{\zeta (rs)}{\zeta(r)}, \frac{\zeta (sr)}{\zeta (r)}
\bigm | 0\ne r\in R \right \} >0.
$$
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 $S=\{s^i|i=1,2,\dots \}$.
Abstract
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 $S=\{s^i|i=1,2,\dots \}$.
Contents
- On polytopes with regular faces in the Lobachesvkij space. (Russian)
- On the cardinality of formula bases and absence of finite approximation with respect to completeness in propositional provability logic. (Russian)
- Description of classically complete chain systems of pseudo-Boolean functions. (Russian)
- Fibre partitions of the Lobachevskij space by limit regular polytopes. (Russian)
- Regular and strongly regular rings with respect to a right ideal. (Russian)
- On a property of groups of type $G^\#$. (Russian)
- Nilpotent linear extensions of minimal groups of transformations. (Russian)
- Topological classification of quadratic systems with a unique third-order singular point. (English)
- On the extension of a pseudonorm of a ring to its field of quotients. (Russian)
- On a family of generalized Hermitian splines. (Russian)
- Waiting for an ape to type a poem. (English)
- Lattices of submodules and Galois connections. (English)
- Centre and multiplication groups of quasigroups. (English)
- Finite projection, reversibility and queueing systems. (English)
