Two-sided residuation in the lattice ordered monoid of topologizing filters 

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Chapter 2 Two-sided residuation in the lattice ordered monoid of topologizing filters

Characterizations of two-sided residuation in Fil RR  for an arbitrary ring

In this chapter we exhibit a number of equivalent conditions which characterize, in module theoretic terms, when the right residual of a pair of elements in [Fil RR]du exists.
A submodule U of M Mod-R is called a hereditary pretorsion submodule of M if U = T (M) for some hereditary pretorsion class T of Mod-R, or equivalently, U = TF(M) for some F Fil RR.
Theorem 2.1 Let F and G be right topologizing filters on a ring R with associated hereditary pretorsion classes TF and TG. Then the following statements are equivalent:

  • The right residual G−1F of F by G exists;
  • There exists a subgenerator M of TF such that the family of all TG-dense hereditary pretorsion submodules of M has a smallest member;
  • For every subgenerator M of TF, the family of all TG-dense hereditary pretorsion submodules of M has a smallest member.Proof. (c)(b) is obvious.
    (b)(a) Let M be a subgenertaor for TF satisfying (b). Denote by MG the smallest TG-dense hereditary pretorsion submodule of M. Put H = FSHC{MG}. We shall demonstrate that H = G−1F. Consider the short exact sequence  0MGM M/MG0.
    Since MG  TH  and M/MG  TG  (because MG  is TG-dense in M), M TG:H.  Since M is a subgenerator for TF, TF TG:H, whence F G : H.
    Now suppose F G : H with H Fil RR. Since M TF TG:H′ , we must have M/TH′ (M) = TG:H′ (M)/TH′ (M) = TG(M/TH′ (M)) TG, so TH′ (M) is a TG-dense submodule of M; TH′ (M) is also, quite obviously, a hereditary pretorsion submodule of M. It follows from the minimality of MG that MG TH′ (M), so MG TH′ . Since H is the smallest member of Fil RR whose associated hereditary pretorsion class contains MG, we must have H H. We conclude that H = G−1F.
    (a)(c) Let M be an arbitrary subgenerator for TF. Put H = G−1F. Note that G : H F. Consider the short exact sequence

    • → TH(M)MM/TH(M)

    Since G : H F, M TF TG:H. It follows that M/TH(M) = TG:H(M)/TH(M) = TG(M/TH(M)) TG, so TH(M) is a TG-dense hereditary pretorsion submodule of M.
    Let N be an arbitrary TG-dense hereditary pretorsion submodule of M. Then N = TH′ (M) for some H Fil RR.
    Consider the short exact sequence
    0NM M/N0

Since N TH′  and M/N TG, we must have M TG:H′ , whence TF = SHC{M} TG:H′ . This implies F G : H, so H = G−1F H and TH(M) TH′ (M) = N. We conclude that TH(M) is the smallest TG-dense hereditary pretorsion submodule of M.
If N M Mod-R, standard theory tells us that the canonical epimorphism πN : M M/N induces a lattice isomorphism πN [] from the set of all submodules of M containing N to the set of all submodules of M/N, and that the mapping πN−1[] constitutes an inverse isomorphism.
If G Fil RR, then the maps πN [] and πN−1[] restrict to mutually inverse bijections between the sets LG[N, M] of all TG-dense submodules of M containing N, and LG[M/N] comprising all TG-dense submodules of M/N. To see this, observe that if N L M, then (M/N)/(L/N) M/L,
thus L will be TG-dense in M if and only if L/N = πN [L] is TG-dense in M/N.
We make use of these rudimentary observations, and adopt the notation used, in the next result.
Corollary 2.2 Let F and G be right topologizing filters on a ring R. Suppose F is compact so that F = FSHC{R/A} for some A RR. The following statements are equivalent:

  • The right residual G−1F of F by G exists;
  • The family of all TG-dense hereditary pretorsion submodule of R/A has a smallest member;
  • The family {B G : B A and B/A is a hereditary pretorsion submodules of R/A} has a smallest member.

Proof. Inasmuch as R/A is a subgenerator for TF, (a)(b) follows from Theorem 2.1 ((a)(c)), whilst (b)(a) is a consequence of Theorem 2.1 ((b)(a)).
(b)(c) Put X = {B G : B A and B/A is a hereditary pretorsion submodule of R/A} and let be the family of all TG-dense hereditary pretorsion submodules of R/A. Let πA : RR R/A be the canonical epimorphism. Observe that B X LG[A, RR] if and only if πA[B] Y LG[R/A]. This means that the map πA[] restricts to a lattice isomorphism from X to Y . The set X will thus possesses a smallest element, that is Statement (c) will hold, precisely if the set Y possess a smallest element, that is, Statement (b) holds.
Theorem 2.3 Let F, G be right topologizing filters on a ring R with associated hereditary pretorsion classes TF and TG. The following statements are equivalent:

    • The right residual G−1F of F by G exists for all F Fil RR satisfying F F;
    • For each M TF the family of all TG-dense hereditary pretorsion submodules of M has a smallest member;
    • For each finitely generated M TF the family of all TG-dense hereditary pretorsion submodules of M has a smallest member;
    • For each cyclic module M TF the family of all TG-dense hereditary pretorsion submodules of M has a smallest member;
    • For each A F the family of all TG-dense hereditary pretorsion submodules of R/A has a smallest member;
    • For each A F the family {B G : B A and B/A is a hereditary pretorsion submodule of R/A} has a smallest member.

Proof. (a)(b) Take M TF and put F = FSHC{M}. Since M is a subgenerator for TF′ it follows from Theorem 2.1 ((a)(c)), that the family of all TG-dense hereditary pretorsion submodules of has a smallest member, as required.
(b)(c)(d) is obvious.
(d)(e) is an immediate consequence of the fact that for a right R-module M, M TF is cyclic if and only if M R/A for some A   F (e)(f) Take A F. Put X = {B G : B A and B/A is a hereditary pretorsion submodule of R/A} and let Y be the family of all TG-dense hereditary pretorsion submodules of R/A. In the proof of Corollary 2.2 it was noted that X will have a smallest member precisely when Y does. The equivalence of (e) and (f) follows.
(e)(a) Note first that if (e) holds in respect of F, G Fil RR, then it will hold for F, G where F is any member of Fil RR satisfying F F. It thus suffices to verify that (e) implies the existence of the right residual G−1F only.

Right fully bounded noetherian rings

In Remark 1.15 we noted that if R is a commutative noetherian ring, then Fil RR is commutative and from this can be drawn the easy inference that [Fil RR]du is two-sided residuated, for [Fil RR]du is left residuated for all rings R. Inasmuch as right fully bounded noetherian rings (these will be defined below) are a natural generalization of commutative noetherian rings and share many properties with their progenitor, it is natural to ask what properties relating to Fil RR carry across from commutative noetherian rings to fully bounded noetherian rings. Commutativity of Fil RR is too much to expect, however, for even in right artinain rings, and such rings are right fully bounded noetherian, ideal multiplication need not commute and since Id R embeds in Fil RR (Theorem 1.14) this would imply the failure of commutativity in the larger Fil RR. A more insightful question is to ask whether the weaker two-sided residuation property holds for [Fil RR]du in all right fully bounded noetherian rings R. The main theorem (Theorem 2.12) in this section answers this question in the affirmative.
Recall that a ring R is said to be right bounded if every essential right ideal of R contains a nonzero two-sided ideals that is essential in RR. If R/P is right bounded for all prime ideals P of R, we say that R is right fully bounded.

Declaration 
Dedication 
Acknowledgements 
Abstract 
Introduction 
Convention on the numbering of results 
1 Preliminaries 
1.1 Set theoretic conventions
1.2 Rings and modules
1.3 Lattice ordered monoids
1.4 Congruence relations on the lattice ordered monoids
1.5 Torsion Theory – hereditary pretorsion classes
1.6 Topologizing filters
1.7 Change of rings
2 Two-sided residuation in the lattice ordered monoid of topologizing filters 
2.1 Characterizations of two-sided residuation in FilRR for an arbitrary ring
2.2 Right fully bounded noetherian rings
3 Topologizing filters in commutative rings 
3.1 Two-sided residuation in FilRR
3.2 Semiartinian rings
3.3 A class of examples of commutative semiartinian rings
3.4 Localization in commutative rings
3.5 Topologizing filters on the ring of fractions RS−1
3.6 An application of congruences on FilR
4 Valuation domains 
4.1 Illustrative examples
5 Open problems and planned future work 
Bibliography 
Nomenclature 
Index
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