nLab
topologizing filter

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

   $$(I:r):=\{s\in R\mid \mathrm{rs}\in I\}$$(I:r) := \{s\in R \,|\, rs\in I\}

   is in $I$.

2. if $J\in F$ and $(I:r)\in I$ 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