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 R_R  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 R_R]^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 = T_F(M) for some F ∈ Fil R_R.
Theorem 2.1 Let F and G be right topologizing filters on a ring R with associated hereditary pretorsion classes T_F and T_G. Then the following statements are equivalent:

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

  Since G : H ⊇ F, M ∈ T_F ⊆ T_G:H. It follows that M/T_H(M) = T_G:H(M)/T_H(M) = T_G(M/T_H(M)) ∈ T_G, so T_H(M) is a T_G-dense hereditary pretorsion submodule of M.
  Let N be an arbitrary T_G-dense hereditary pretorsion submodule of M. Then N = T_H′ (M) for some H^′ ∈ Fil R_R.
  Consider the short exact sequence
  0→N→M →M/N→0

Since N ∈ T_H′  and M/N ∈ T_G, we must have M ∈ T_G:H′ , whence T_F = SHC{M} ⊆ T_G:H′ . This implies F ⊆ G : H^′, so H = G⁻¹F ⊆ H^′ and T_H(M) ⊆ T_H′ (M) = N. We conclude that T_H(M) is the smallest T_G-dense hereditary pretorsion submodule of M.
If N ≤ M ∈ Mod-R, standard theory tells us that the canonical epimorphism π_N : M → M/N induces a lattice isomorphism π_N [₋] from the set of all submodules of M containing N to the set of all submodules of M/N, and that the mapping π_N⁻¹[₋] constitutes an inverse isomorphism.
If G ∈ Fil R_R, then the maps π_N [₋] and π_N⁻¹[₋] restrict to mutually inverse bijections between the sets L_G[N, M] of all T_G-dense submodules of M containing N, and L_G[M/N] comprising all T_G-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 T_G-dense in M if and only if L/N = π_N [L] is T_G-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 = F_SHC{R/A} for some A ≤ R_R. The following statements are equivalent:

• The right residual G⁻¹F of F by G exists;
• The family of all T_G-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 T_F, (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 T_G-dense hereditary pretorsion submodules of R/A. Let π_A : R_R → R/A be the canonical epimorphism. Observe that B ∈ X ⊆ L_G[A, R_R] if and only if π_A[B] ∈ Y ⊆ L_G[R/A]. This means that the map π_A[₋] 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 T_F and T_G. The following statements are equivalent:

• • The right residual G⁻¹F^′ of F^′ by G exists for all F^′ ∈ Fil R_R satisfying F^′ ⊆ F;
  • For each M ∈ T_F the family of all T_G-dense hereditary pretorsion submodules of M has a smallest member;
  • For each finitely generated M ∈ T_F the family of all T_G-dense hereditary pretorsion submodules of M has a smallest member;
  • For each cyclic module M ∈ T_F the family of all T_G-dense hereditary pretorsion submodules of M has a smallest member;
  • For each A ∈ F the family of all T_G-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 ∈ T_F and put F^′ = F_SHC{M}. Since M is a subgenerator for T_F′ it follows from Theorem 2.1 ((a)⇒(c)), that the family of all T_G-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 ∈ T_F 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 T_G-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 R_R, then it will hold for F^′, G where F^′ is any member of Fil R_R 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 R_R is commutative and from this can be drawn the easy inference that [Fil R_R]^du is two-sided residuated, for [Fil R_R]^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 R_R carry across from commutative noetherian rings to fully bounded noetherian rings. Commutativity of Fil R_R 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 R_R (Theorem 1.14) this would imply the failure of commutativity in the larger Fil R_R. A more insightful question is to ask whether the weaker two-sided residuation property holds for [Fil R_R]^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 R_R. 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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