nLab
topologizing filter

Topologizing filters

Definition

Let RR 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 FF such that RFR\in F; if II,JJ are right ideals, IFI\in F and IJI\subset J then JFJ\in F; and if I,JFI,J\in F then IJFI\cap J\in F.

A filter of ideals in RR is topologizing if

  1. it is uniform, i.e. for any IFI\in F and rRr\in R, the right ideal

    (I:r):={sR|rsI} (I:r) := \{s\in R \,|\, rs\in I\}

    is in II.

  2. if JFJ\in F and (I:r)I(I:r)\in I for all rJr\in J then IFI\in F.

Properties

The term topologizing is explained by the following statement:

Proposition

A topologizing set of right ideals in a ring RR is a basis of neighborhoods of 00 for a topology on RR.

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)