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Applications of reduced and coreduced modules II: Radicality of the functor $\text{Hom}_R(R/I, -)$

This paper establishes necessary and sufficient conditions for the functor $\text{Hom}_R(R/I, -)$ to be a radical on an Abelian full subcategory of $R$ -modules in terms of $I$ -reduced modules, while also using $I$ -reduced and $I$ -coreduced modules to generalize Jans' correspondence and construct a new radical class of rings.

Original authors: David Ssevviiri

Published 2026-06-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: David Ssevviiri

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a librarian trying to organize a massive, chaotic library. In this library, the "books" are mathematical objects called modules, and the "shelves" are defined by rules set by a specific "ideal" (a special collection of numbers or rules) called I.

The paper by David Ssevviiri is about finding the right way to sort these books so that a specific sorting machine (a mathematical tool called a functor) works perfectly.

Here is the breakdown of the paper's journey, using everyday analogies:

1. The Problem: A Broken Sorting Machine

The author starts with a machine called Hom(R/I, −). Think of this machine as a scanner that looks at a book and asks: "Does this book belong to the section defined by rule I?"

The Goal: The author wants this machine to be a "radical." In math-speak, a "radical" is a perfect, reliable filter. If you run a book through it, it cleanly separates the "bad" parts from the "good" parts, and if you run the result through it again, nothing changes.
The Issue: In the general library (the category of all modules), this machine is broken. It doesn't always give a clean result. Sometimes it leaves a mess behind, or it fails to stop changing the book when you run it twice.
Why? The machine fails because the library contains "messy" books. Specifically, it contains books where a rule $I$ applied twice ( $I^2$ ) kills a page, but applying it once ( $I$ ) didn't. These are "confused" books that don't follow the simple logic the machine needs.

2. The Solution: Two Special Rooms (Reduced and Coreduced)

The author realizes that if we restrict the machine to work only in two specific, well-organized rooms, it works perfectly again.

Room A: The "Reduced" Room ( $I$ -reduced modules)
- The Rule: In this room, if a rule $I$ applied twice makes a page vanish, then applying it once must have already made it vanish. There are no "half-killed" pages.
- The Result: When the machine works in this room, it becomes a perfect radical. It sorts things cleanly.
- The Analogy: Imagine a room where every book is either fully intact or fully destroyed. There are no "partially torn" books. In this room, the scanner works perfectly.
Room B: The "Coreduced" Room ( $I$ -coreduced modules)
- The Rule: In this room, the books are structured so that applying the rule $I$ twice is exactly the same as applying it once. The structure is stable.
- The Result: Here, a different machine (one that looks at what is left after applying the rule) also works perfectly.

3. The Big Discovery: Fixing Jans' Correspondence

There is a famous old rule in mathematics called Jans' Correspondence.

The Old Rule: It said, "If your rule $I$ is 'idempotent' (meaning applying it twice is the same as applying it once, like a light switch that is either ON or OFF), then there is a perfect one-to-one match between the rule and the books it filters."
The Problem: Many rules in math are not perfect switches. They are dimmer switches. The old rule broke down for these dimmer switches.
The Paper's Fix: The author shows that even if the rule isn't a perfect switch (it's just "flat," a technical term meaning it plays nicely with other rules), we can still get a perfect match!
- We just have to split the library into the two special rooms mentioned above (Reduced and Coreduced).
- In the Reduced Room, the machine acts as a "Torsion" filter (finding the bad parts).
- In the Coreduced Room, the machine acts as a "Torsion-free" filter (finding the good parts).
- The Magic: The same group of books (those where the rule $I$ kills them completely) sits in the middle, acting as the "bad" group in one room and the "good" group in the other. This restores the perfect correspondence that Jans found, but for a much wider variety of rules.

4. What This Means for Rings (The "Radical Class")

The author takes these findings about books (modules) and applies them to the library buildings themselves (rings).

They define a new "Radical Class" of rings. Think of this as a "Club" of rings that have a specific property: if you apply the rule $I$ to them, they vanish completely.
The paper proves that this Club is a valid "Radical Class," meaning it follows all the strict rules of mathematical club membership (if you join, your sub-clubs join; if you leave, you leave cleanly).

5. The Spectral Sequences (The "X-Ray Machines")

Finally, the author uses these special rooms to look at "Spectral Sequences."

The Analogy: Imagine an X-ray machine that takes a picture of a book, then takes a picture of that picture, and so on, to reveal hidden layers. Usually, these pictures get blurry or messy.
The Result: In the special "Reduced" and "Coreduced" rooms (specifically in rings that are "von Neumann regular," a type of very orderly ring), the X-ray machine becomes incredibly simple.
- It takes one picture, and that's it. The picture doesn't change.
- The author calculates exactly what these pictures look like, showing that in these orderly rooms, the complex layers of math collapse into a single, clear image.

Summary

The paper is a guide on how to fix a broken mathematical sorting machine.

The Machine: A tool that checks if objects belong to a specific category.
The Breakdown: It fails in the general world because the world is too messy.
The Fix: Move the machine into two specific, tidy rooms ("Reduced" and "Coreduced").
The Payoff: In these rooms, the machine works perfectly, restoring a famous mathematical connection (Jans' Correspondence) that was thought to be broken, and allowing for clearer "X-ray" views of complex mathematical structures.

The author does not claim this has immediate medical or industrial uses; it is purely a theoretical advancement in how mathematicians understand the structure of rings and modules.

Technical Summary: Applications of Reduced and Coreduced Modules II: Radicality of the Functor HomR(R/I, −)

Problem Statement
The paper addresses a fundamental limitation in the theory of radicals and torsion theories within the category of modules over a commutative unital ring $R$ . Specifically, it investigates the conditions under which the functor $\text{Hom}_R(R/I, -)$ , which is naturally isomorphic to the submodule $(0 :_M I)$ , constitutes a radical on the category of $R$ -modules ( $R\text{-Mod}$ ).

While Jans' correspondence establishes a one-to-one relationship between idempotent ideals $I$ and torsion-torsionfree (TTF) classes, this correspondence fails for non-idempotent ideals. The core problem identified is that for non-idempotent ideals, the functor $\text{Hom}_R(R/I, -)$ is a left exact preradical but generally fails to be a radical (i.e., it does not satisfy $\gamma(M/\gamma(M)) = 0$ ). Consequently, the standard torsion-theoretic framework does not apply directly to non-idempotent ideals in the full module category. The paper seeks to determine necessary and sufficient conditions for this functor to be a radical and to generalize Jans' correspondence to settings where the ideal $I$ is not necessarily idempotent.

Methodology
The author employs a categorical and module-theoretic approach, utilizing the concepts of $I$ -reduced and $I$ -coreduced modules to construct suitable subcategories where the desired properties hold.

Definitions and Subcategories:
- An $R$ -module $M$ is $I$ -reduced if $I^2m = 0 \implies Im = 0$ for all $m \in M$ .
- An $R$ -module $M$ is $I$ -coreduced if $I^2M = IM$ .
- The paper defines $\mathcal{A}_I$ and $\mathcal{B}_I$ as Serre subcategories of $R\text{-Mod}$ consisting of $I$ -reduced and $I$ -coreduced modules, respectively.
Radicality Analysis:
The paper analyzes the functor $\text{Hom}_R(R/I, -)$ restricted to these Serre subcategories. It utilizes the chain of submodules $(0 :_M I) \subseteq (0 :_M I^2) \subseteq \dots$ to demonstrate that the functor becomes a radical precisely when the modules in the category are $I$ -reduced. Similarly, the functor $\delta_I(M) = IM$ is shown to be an idempotent radical on $\mathcal{B}_I$ if and only if the modules are $I$ -coreduced.
Torsion Theory Construction:
Assuming $I$ is a flat $R$ -module, the author constructs two distinct torsion theories:
- A hereditary torsion theory $(\mathcal{T}, \mathcal{F})$ on the subcategory $\mathcal{A}_I$ .
- A torsion theory $(\mathcal{T}, \mathcal{T})$ on the subcategory $\mathcal{B}_I$ .
  Here, the class $\mathcal{T} = \{M \in \mathcal{A}_I \mid (0:_M I) = M\} = \{M \in \mathcal{B}_I \mid IM = M\}$ serves simultaneously as the torsion class in the first theory and the torsionfree class in the second.
Spectral Sequences:
The paper applies these module-theoretic results to compute Grothendieck spectral sequences associated with the $I$ -torsion functor $\Gamma_I$ and the $I$ -adic completion functor $\Lambda_I$ , particularly in the context of von Neumann regular rings and Artinian rings.

Key Contributions and Results

Characterization of Radicality: The paper proves that $\text{Hom}_R(R/I, -)$ is a radical on the full category $R\text{-Mod}$ if and only if $I$ is an idempotent ideal (Theorem 3.3). More generally, it establishes that restricting the functor to the Serre subcategory $\mathcal{A}_I$ of $I$ -reduced modules restores radicality (Proposition 2.2).
Generalization of Jans' Correspondence: Theorem 3.2 generalizes Jans' correspondence. While Jans' correspondence requires $I$ to be idempotent to define a TTF class in $R\text{-Mod}$ , this paper shows that if $I$ is merely flat, one obtains two interacting torsion theories on the subcategories $\mathcal{A}_I$ and $\mathcal{B}_I$ . The class of modules annihilated by $I$ (or satisfying $IM=M$) acts as a "bridge" class, being torsion in one theory and torsionfree in the other.
Radical Class of Rings: The paper defines a radical class of rings $\Psi_I = \{S \in \mathcal{C}_R : IS = 0\}$ for an idempotent ideal $I$ , where $\mathcal{C}_R$ is the category of rings with compatible $R$ -module structures. It proves $\Psi_I$ is a radical class and establishes the isomorphism $\text{Hom}_R(R/I, R) \cong \Psi_I(R) = \Gamma_I(R)$ (Theorem 5.1, Corollary 5.2).
Improvement of Literature: The paper refines existing results regarding weakly proregular ideals. It proves that any finitely generated idempotent ideal is weakly proregular, dropping the "strongly" adjective required in previous literature (Proposition 4.1).
Spectral Sequence Computations: For von Neumann regular rings, the paper demonstrates that the Grothendieck spectral sequences for local cohomology and local homology collapse trivially at degree zero, recovering the functors $\Gamma_I$ and $\Lambda_I$ directly (Propositions 6.1–6.3).

Significance and Claims
The paper claims that the "failure of Jans' correspondence for non-idempotent ideals is exactly the failure of $\text{Hom}_R(R/I, -)$ to be a radical on $R\text{-Mod}$ ." By restricting attention to suitable categories of $I$ -reduced and $I$ -coreduced modules, the paper argues that radicality is restored, yielding generalized torsion-theoretic correspondences.

The conceptual contribution is framed as providing a "natural setting" for generalizing Jans' correspondence and constructing a new radical class of rings. The results offer a mechanism to recover radical-theoretic behavior in situations where $I$ is not idempotent and higher local cohomology is non-trivial. The paper does not propose new experimental applications but rather extends the theoretical framework of radicals, torsion theories, and spectral sequences within commutative algebra.