Similar submodules of projective modules

The Big Picture: Organizing a Messy Library

Imagine you have a massive, complex library (this is your Module). Inside this library, there are many different sections or shelves (these are Submodules). Some of these shelves are "maximal," meaning they are the biggest possible sections you can have without the whole library collapsing into chaos.

The author, Alborz Azarang, is asking a very specific question: How many of these "biggest shelves" can look exactly the same from the outside, even if they are in different parts of the library?

In math, when two things look the same structurally, we call them similar. The paper introduces a new way to group these shelves based on their "similarity."

Key Concepts Translated

1. The Library and the Manager (Modules and Endomorphism Rings)

The Library ( $M$ ): This is the object we are studying. It's a collection of items with specific rules on how they can be moved or rearranged.
The Manager ( $End_R(M)$ ): Imagine a manager who has a master key and a list of every possible way to rearrange the books in the library. This manager's rulebook is called the Endomorphism Ring.
The Connection: The paper discovers a secret tunnel between the shelves in the library and the rules in the manager's book. If you find a specific "maximal shelf" in the library, it corresponds to a specific "maximal rule" in the manager's book.

2. The "Look-Alike" Shelves (Similarity)

In the old days, mathematicians knew how to tell if two rules (ideals) were similar. Azarang figured out how to tell if two shelves (submodules) are similar.

The Analogy: Imagine two different rooms in a hotel. Even if they are in different wings, if they have the exact same layout, furniture, and view, they are "similar."
The Discovery: If a shelf is not "fully invariant" (meaning the manager can't just leave it alone; they have to move things around it), then it cannot be unique. It must have look-alikes.

3. The "Crowded Room" Theorem (The Main Result)

This is the paper's biggest punchline.

The Scenario: You find a "maximal shelf" in your library.
The Rule: If this shelf is special (fully invariant), it might be the only one of its kind. But, if it's not special, it belongs to a "clique."
The Result: If a shelf isn't special, it must have at least three look-alikes (including itself). In fact, the number of look-alikes depends on the "eigenring" (a fancy math term for a specific type of number system associated with that shelf).
Simple Takeaway: You can't have just one or two of these special, non-invariant shelves. They come in packs. If you find one, you've found a whole group.

4. The "Perfect Library" (Faithfully Projective Modules)

Some libraries are "perfect" (Faithfully Projective). In these libraries, the connection between the shelves and the manager's rules is a perfect 1-to-1 match.

The Analogy: It's like a library where every single shelf has a unique, corresponding rule in the manager's book, and vice versa.
The Consequence: If the manager's book is finite (has a limited number of pages), then the library itself must be finite (it has a limited number of books). This allows mathematicians to count things in the library by just counting things in the manager's book.

5. The "Infinite Hotel" (Matrix Rings)

The paper ends by applying these rules to a specific type of structure called a Matrix Ring (think of a grid of numbers, like a spreadsheet).

The Problem: If you have an infinite division ring (an infinite set of numbers where you can divide by anything except zero), and you build a matrix ring larger than 1x1 (like a 2x2 grid), how many "maximal shelves" does it have?
The Answer: Infinitely many.
The Metaphor: Imagine a hotel with an infinite number of rooms. If you try to group the rooms into "maximal sections" that aren't the whole hotel, you will find that you can't stop counting. There are infinitely many ways to slice the hotel that aren't just "the whole thing."
Why it matters: This solves an old puzzle in algebra. It proves that for these infinite structures, you can never have just a few "special" rules; you always have an endless supply of them.

Summary of the "Story"

The Setup: Mathematicians wanted to know how to count the "biggest possible parts" of a mathematical object.
The Innovation: Azarang created a new way to say when two parts are "twins" (similar).
The Discovery: If a part isn't "locked down" (fully invariant), it can't be lonely. It must have at least two twins. In fact, the number of twins is huge if the underlying math system is big.
The Application: This rule proves that if you take an infinite set of numbers and arrange them in a grid (matrix), you will always find an infinite number of unique "maximal" ways to organize them.

In a nutshell: The paper shows that in the world of abstract algebra, uniqueness is rare. If something isn't perfectly rigid, it comes in a crowd. And if your numbers are infinite, that crowd is endless.

1. Problem Statement and Motivation

The paper addresses a gap in the structural theory of modules and rings regarding the relationship between the lattice of submodules of a projective module and the ideal structure of its endomorphism ring.

Context: It is well-established that a ring $R$ has a unique maximal right ideal if and only if it has a unique maximal left ideal (i.e., $R$ is a local ring). Ware previously investigated the module-theoretic analogue: if a projective module $P$ has a unique maximal submodule, is $P$ a local module? While settled for specific classes (commutative, left Noetherian) and recently in full generality by Sato, a systematic framework linking the collection of maximal submodules to the ideal structure of the endomorphism ring was missing.
The Gap: The classical theory of similarity for right ideals (where $I \sim J$ if $R/I \cong R/J$ ) is rich and connects to eigenrings and idealizers. However, this theory had not been extended to submodules of general modules in a way that preserves its structural power, particularly for projective modules.
Objective: To define a similarity relation for submodules, establish a correspondence between maximal submodules of a projective module and maximal right ideals of its endomorphism ring, and derive lower bounds on the number of maximal submodules (and non-two-sided maximal ideals in rings).

2. Methodology and Definitions

The author employs homological algebra, module theory, and ring theory techniques, specifically focusing on:

Similarity of Submodules:
- Definition: Two submodules $N, N' \subseteq M$ are similar ( $N \sim N'$ ) if the quotient modules are isomorphic: $M/N \cong M/N'$ .
- Eigenring: For a submodule $N$ , the idealizer $I(N) = \{ \phi \in \text{End}_R(M) \mid \phi(N) \subseteq N \}$ and the eigenring $E(N) = I(N) / \text{Hom}_R(M, N)$ .
- Characterization: For a projective module $M$ , $N \sim N'$ if and only if there exists an endomorphism $\beta$ such that $N' + \beta(M) = M$ and $N = (N' : \beta) = \{m \in M \mid \beta(m) \in N'\}$ .
Key Module Classes:
- Retractable/Slightly Compressible: Modules where $\text{Hom}_R(M, N) \neq 0$ for all non-zero submodules $N$ .
- Faithfully Projective: Modules where the functor $\text{Hom}_R(M, -)$ is faithfully exact. This implies $M$ is a generator for the category of right $R$ -modules.
Correspondence Map: The paper utilizes the map $\Phi: N \mapsto \text{Hom}_R(M, N)$ , which maps submodules of $M$ to right ideals of $E = \text{End}_R(M)$ .

3. Key Contributions and Results

A. Similarity and the Abundance of Maximal Submodules

The central result establishes a sharp lower bound for the number of maximal submodules based on the eigenring.

Theorem 3.2: Let $M$ $M$ be an $R$ $R$ -module and $N \in \text{Max}(M)$ $N \in Max (M)$ . Then either:
1. $N$ is fully invariant (i.e., $I(N) = \text{End}_R(M)$ ), or
2. $N$ is similar to at least $1 + |E(N)|$ distinct maximal submodules.
Corollary: If $N$ is not fully invariant, then $|\text{Max}(M)| \ge 3$ .
Algebraic Extension: If $R$ is a $K$ -algebra over a field $K$ , then $|E(N)| \ge |K|$ . Thus, if $N$ is not fully invariant, $|\text{Max}(M)| \ge 1 + |K|$ .

B. Correspondence with Endomorphism Ring Ideals

For projective modules, the author constructs a canonical injection between the set of maximal submodules of $M$ and the set of maximal right ideals of $E = \text{End}_R(M)$ .

Theorem 3.5: If $M$ is projective, the map $N \mapsto \text{Hom}_R(M, N)$ is a one-to-one function from $\text{Max}(M)$ into $\text{Max}_r(E)$ .
Implication: This provides a new proof that a projective module $M$ is local (has a unique maximal submodule) if and only if its endomorphism ring $E$ is a local ring.

C. Finiteness and Decomposition

The paper links the length of the module to the length of its endomorphism ring, specifically for faithfully projective modules.

Theorem 3.7: If $M$ is faithfully projective and $E$ is right Artinian, then $M$ has finite length and decomposes into a direct sum of local (strongly indecomposable) summands: $M \cong \bigoplus (e_i R)^{n_i}$ .
Theorem 3.8:
- If $M$ has finite length, then $E$ has finite length with $\ell(E_E) \le \ell(M_R)$ .
- If $M$ is faithfully projective, then $\ell(E_E) = \ell(M_R)$ .
- Conversely, if $\ell(E_E) = \ell(M_R)$ , then $M$ is slightly compressible.

D. Equivalence of Similarity for Faithfully Projective Modules

Corollary 3.15: For a faithfully projective module $M$ , two submodules $N, N'$ are similar if and only if their corresponding right ideals $\text{Hom}_R(M, N)$ and $\text{Hom}_R(M, N')$ are similar in $E$ . (This equivalence fails for general projective modules, as shown by Example 3.12).

4. Applications to Ring Theory

The module-theoretic results are transferred to obtain new bounds on the number of maximal one-sided ideals in rings that are not two-sided.

Lower Bounds on Non-Two-Sided Ideals:
- Theorem 3.21: If $M$ is a maximal right ideal of a ring $T$ that is not a two-sided ideal, then the similarity class $[M]$ has size $|[M]| \ge |I(M)/M| + 1$ .
- Corollary 3.22/3.23: If $T$ is a $K$ -algebra and not right quasi-duo, it possesses at least $|K| + 1$ maximal right ideals that are not two-sided. If $K$ is infinite, $T$ has infinitely many such ideals.
Matrix Rings over Infinite Division Rings:
- Corollary 3.24: If $R$ is an infinite division ring and $n > 1$ , the matrix ring $M_n(R)$ has infinitely many maximal left (and right) ideals.
- Generalization (Corollary 3.28): If a ring $T$ contains $M_n(D)$ as a subring (where $D$ is an infinite division ring), then $T$ has infinitely many maximal left/right ideals.
Structure of Prime/Primitive Rings:
- Corollary 3.26: For a prime ring $R$ with $J(R)=0$ , or a left/right primitive ring, either $R$ is a division ring or it has infinitely many maximal left/right ideals.

5. Significance

Theoretical Unification: The paper successfully extends the classical theory of similar right ideals to submodules of projective modules, creating a bridge between the submodule lattice of a module and the ideal lattice of its endomorphism ring.
Quantitative Insights: It provides concrete lower bounds for the number of maximal submodules, resolving the "abundance" question. It shows that unless a maximal submodule is fully invariant, there must be a large (often infinite) family of similar maximal submodules.
Ring Theory Applications: The results offer powerful tools for analyzing the ideal structure of non-commutative rings, particularly matrix rings and algebras over infinite fields, proving that "pathological" cases (rings with very few maximal one-sided ideals) are extremely restricted (essentially limited to division rings).
Faithful Projectivity: The distinction made for faithfully projective modules (where similarity of submodules is equivalent to similarity of corresponding ideals) clarifies the structural behavior of generators in module categories.

In summary, Azarang's work provides a robust framework for analyzing the "multiplicity" of maximal submodules via similarity, yielding deep structural consequences for both module theory and non-commutative ring theory.