LAUSR

Abstract The order dual $${[{\rm Fil}{R_{R}}]^{{\rm du}}}$$[FilRR]du of the set $${{\rm Fil}{R_{R}}}$$FilRR 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 $${[{\rm Fil}{R_{R}}]^{{\rm du}}}$$[FilRR]du is always left residuated, meaning, for each pair $${\mathfrak{F}, \mathfrak{G} \in {\rm Fil}{R_{R}}}$$F,G∈FilRR there exists a smallest $${\mathfrak{H} \in {\rm Fil}{R_{R}}}$$H∈FilRR such that $${\mathfrak{H}: \mathfrak{G} \supseteq \mathfrak{F}}$$H:G⊇F , but is not, in general, right residuated (there exists a smallest $${\mathfrak {H}}$$H such that $${\mathfrak{G} : \mathfrak{H} \supseteq \mathfrak{F}}$$G:H⊇F). Rings R for which $${[{\rm Fil}{R_{R}}]^{{\rm du}}}$$[FilRR]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 $${\hat{\varphi}_{P}}$$φ^P from $${[{\rm Fil}{R}]^{{\rm du}}}$$[FilR]du to $${[{\rm Fil}{R_{P}}]^{{\rm du}}}$$[FilRP]du where RP denotes the localization of R at P. The kernel $${\equiv_{\hat{\varphi}_{P}}}$$≡φ^P of $${\hat{\varphi}_{P}}$$φ^P is a congruence on $${[{\rm Fil}{R}]^{{\rm du}}}$$[FilR]du whose properties we explore. Defining $${{\rm Rad}({\rm Fil}{R})}$$Rad(FilR) to be the intersection of all congruences $${\equiv_{\hat{\varphi}_{P}}}$$≡φ^P as P ranges through all maximal ideals of R, we show that for commutative VNR rings R, $${{\rm Rad}({\rm Fil}{R})}$$Rad(FilR) 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, $${{\rm Rad}({\rm Fil}{R})}$$Rad(FilR) is trivial whenever $${{\rm Fil}{R}}$$FilR is commutative (meaning, the monoid operation ‘$${:}$$:’ on $${{\rm Fil}{R}}$$FilR is commutative). This yields, for such rings R, a subdirect embedding of $${[{\rm Fil}{R}]^{{\rm du}}}$$[FilR]du into the product of all $${[{\rm Fil}{R_{P}}]^{{\rm du}}}$$[FilRP]du as P ranges through all maximal ideals of R. The theory developed is used to prove that a Prüfer domain R for which $${{\rm Fil}{R}}$$FilR is commutative, is necessarily Dedekind.