# Commutativity in the lattice of topologizing filters of a ring – localization and congruences

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## Abstract

The order dual \({[{\rm Fil}{R_{R}}]^{{\rm du}}}\) of the set \({{\rm Fil}{R_{R}}}\) of all right topologizing filters on a fixed but arbitrary ring *R* is a complete lattice ordered monoid with respect to the (order dual) of inclusion and a monoid operation ‘\({:}\)’ that is, in general, noncommutative. It is known that \({[{\rm Fil}{R_{R}}]^{{\rm du}}}\) is always left residuated, meaning, for each pair \({\mathfrak{F}, \mathfrak{G}
\in {\rm Fil}{R_{R}}}\) there exists a smallest \({\mathfrak{H} \in {\rm Fil}{R_{R}}}\) such that \({\mathfrak{H}: \mathfrak{G} \supseteq \mathfrak{F}}\) , but is not, in general, right residuated (there exists a smallest \({\mathfrak {H}}\) such that \({\mathfrak{G} : \mathfrak{H} \supseteq \mathfrak{F}}\)). Rings *R* for which \({[{\rm Fil}{R_{R}}]^{{\rm du}}}\) is both left and right residuated are shown to satisfy the DCC on left annihilator ideals and possess only finitely many minimal prime ideals.

It is shown that every maximal ideal *P* of a commutative ring *R* gives rise to an onto homomorphism of lattice ordered monoids \({\hat{\varphi}_{P}}\) from \({[{\rm Fil}{R}]^{{\rm du}}}\) to \({[{\rm Fil}{R_{P}}]^{{\rm du}}}\) where *R*_{P} denotes the localization of *R* at *P*. The kernel \({\equiv_{\hat{\varphi}_{P}}}\) of \({\hat{\varphi}_{P}}\) is a congruence on \({[{\rm Fil}{R}]^{{\rm du}}}\) whose properties we explore. Defining \({{\rm Rad}({\rm Fil}{R})}\) to be the intersection of all congruences \({\equiv_{\hat{\varphi}_{P}}}\) as *P* ranges through all maximal ideals of *R*, we show that for commutative VNR rings *R*, \({{\rm Rad}({\rm Fil}{R})}\) is trivial (the identity congruence) precisely if *R* is noetherian (and thus a finite product of fields). It is shown further that for arbitrary commutative rings *R*, \({{\rm Rad}({\rm Fil}{R})}\) is trivial whenever \({{\rm Fil}{R}}\) is commutative (meaning, the monoid operation ‘\({:}\)’ on \({{\rm Fil}{R}}\) is commutative). This yields, for such rings *R*, a subdirect embedding of \({[{\rm Fil}{R}]^{{\rm du}}}\) into the product of all \({[{\rm Fil}{R_{P}}]^{{\rm du}}}\) as *P* ranges through all maximal ideals of *R*. The theory developed is used to prove that a Prüfer domain *R* for which \({{\rm Fil}{R}}\) is commutative, is necessarily Dedekind.

## Key words and phrases

topologizing filter lattice ordered monoid residuated lattice localization congruence Prüfer domain Dedekind domain## Mathematics Subject Classification

primary 16S90 secondary 06F05 13F05 16P50## Preview

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## Notes

### Acknowledgement

The authors thank the referee for a number of suggestions that have significantly improved the paper.

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