# Topologizing filters

## Definition

Let $R$ be a (possibly noncommutative associative) ring (unital or not). The collection of one sided (say right) ideals has a partial order with respect to inclusion; a filter with respect to this partial order is a set of right ideals $F$ such that $R\in F$; if $I$,$J$ are right ideals, $I\in F$ and $I\subset J$ then $J\in F$; and if $I,J\in F$ then $I\cap J\in F$.

A filter of ideals in $R$ is topologizing if

1. it is uniform, i.e. for any $I\in F$ and $r\in R$, the right ideal

$\left(I:r\right):=\left\{s\in R\phantom{\rule{thinmathspace}{0ex}}\mid \phantom{\rule{thinmathspace}{0ex}}\mathrm{rs}\in I\right\}$(I:r) := \{s\in R \,|\, rs\in I\}

is in $I$.

2. if $J\in F$ and $\left(I:r\right)\in I$ for all $r\in J$ then $I\in F$.

## Properties

The term topologizing is explained by the following statement:

###### Proposition

A topologizing set of right ideals in a ring $R$ is a basis of neighborhoods of $0$ for a topology on $R$.

## References

• Carl Faith, Algebra Vol. I, page 520

• Harold Simmons, The semiring of topologizing filters of a ring, doi

Revised on January 26, 2011 17:06:30 by Toby Bartels (64.89.48.241)