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arxiv: 2604.03243 · v1 · submitted 2026-02-12 · 🧮 math.RA · math.RT

Similar submodules of projective modules

Alborz Azarang This is my paper

Pith reviewed 2026-05-16 02:24 UTC · model grok-4.3

classification 🧮 math.RA math.RT
keywords similarity relationmaximal submodulesprojective moduleseigenringendomorphism ringfinite lengthmatrix ringsone-sided ideals
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The pith

If a maximal submodule of a faithfully projective module is not fully invariant, then it is similar to at least two other distinct maximal submodules.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a new similarity relation among submodules of a module that generalizes the classical similarity for right ideals. It then proves a lower bound on the number of maximal submodules in faithfully projective modules: any maximal submodule is either fully invariant or similar to a collection whose size is determined by the eigenring of the submodule, yielding at least three maximal submodules in the non-invariant case. Additional results establish a canonical one-to-one map between maximal submodules of the module and maximal right ideals of its endomorphism ring. The paper further relates the finite length of the module to that of its endomorphism ring and derives consequences for the ideal structure of matrix rings over infinite algebras.

Core claim

If N is a maximal submodule of a (faithfully) projective module M, then either N is fully invariant or N is similar to at least 1+|S| distinct maximal submodules, where S is the eigenring of N; in particular |Max(M)| >= 1+|S| >= 3 in the latter case. For projective modules there is a canonical one-to-one map from Max(M) into Max_r(End_R(M)). When M is faithfully projective and End_R(M) is right Artinian, M has finite length and decomposes into a direct sum of local summands. Conversely, if M is a projective right R-module with finite length, then E_E has finite length with ℓ(E_E) ≤ ℓ(M_R); moreover, if M is faithfully projective then ℓ(E_E) = ℓ(M_R).

What carries the argument

The similarity relation between submodules of M extending classical similarity for right ideals, together with the eigenring S of a submodule N, used to produce lower bounds on the total number of maximal submodules.

If this is right

  • A canonical one-to-one map exists from the set of maximal submodules of a projective module M to the set of maximal right ideals of the endomorphism ring of M.
  • If M is faithfully projective with right Artinian endomorphism ring, then M has finite length and is a direct sum of local summands.
  • For a projective module M of finite length, the endomorphism ring has finite length bounded above by the length of M, with equality if M is faithfully projective.
  • Equality of lengths implies that M is slightly compressible.
  • These yield explicit lower bounds on the number of maximal one-sided ideals that are not two-sided in matrix rings over infinite algebras.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The similarity classes could be used to classify projective modules that possess the smallest possible number of maximal submodules.
  • The length comparison between module and endomorphism ring may serve as a test for faithful projectivity in broader module categories.
  • The matrix-ring consequences suggest new restrictions on the non-two-sided maximal ideals of algebras that are infinite-dimensional over their base rings.

Load-bearing premise

The introduced similarity relation on submodules satisfies transitivity and counting properties that let the size of the eigenring S determine a lower bound on the number of distinct similar maximal submodules.

What would settle it

A faithfully projective module containing a maximal submodule that is not fully invariant but belongs to a similarity class smaller than 1 plus the order of its eigenring would falsify the lower bound.

read the original abstract

We introduce a similarity relation between submodules of a module $M$ over a ring $R$, extending the classical notion of similarity for right ideals. Focusing on (faithfully) projective modules, we establish a sharp lower bound for the number of maximal submodules: if $N$ is a maximal submodule of $M$, then either $N$ is fully invariant or $N$ is similar to at least $1+|S|$ distinct maximal submodules, where $S$ is the eigenring of $N$; in particular, $|{\rm Max}(M)|\geq 1+|S|\geq 3$ in the latter case. For projective modules, we construct a canonical one-to-one map from ${\rm Max}(M)$ into ${\rm Max}_r({\rm End}_R(M))$. When $M$ is faithfully projective and ${\rm End}_R(M)$ is right Artinian, we prove that $M$ has finite length and decomposes into a direct sum of local summands. Conversely, if $M$ is a projective right $R$-module with finite length, then $E_E$ has finite length with $\ell(E_E)\leq \ell(M_R)$; moreover, if $M$ is a faithfully projective $R$-module, then $\ell(E_E)=\ell(M_R)$; conversely, if $\ell(E_E)=\ell(M_R)$ holds, then $M$ is slightly compressible. These results are applied to obtain lower bounds on the number of maximal one-sided ideals that are not two-sided, with explicit consequences for matrix rings over infinite algebras.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces a similarity relation ~ between submodules of a module M over a ring R, extending the classical notion for right ideals. For (faithfully) projective modules, it establishes that if N is maximal then either N is fully invariant or N is similar to at least 1+|S| distinct maximal submodules (S the eigenring of N), yielding |Max(M)| >= 3 in the latter case. A canonical injection Max(M) -> Max_r(End_R(M)) is constructed. When M is faithfully projective and End_R(M) is right Artinian, M has finite length and decomposes as a direct sum of local summands. For finite-length projective M, length inequalities hold between M and its endomorphism ring E, with equality implying M is slightly compressible. These yield lower bounds on non-two-sided maximal one-sided ideals in matrix rings over infinite algebras.

Significance. If the similarity relation is transitive and the eigenring counting argument is injective, the work supplies new, explicit lower bounds on the number of maximal submodules of projective modules and connects them to Artinian endomorphism rings and length comparisons. The canonical map to maximal right ideals of the endomorphism ring and the applications to matrix rings over infinite algebras provide concrete tools for noncommutative ring theory.

major comments (3)
  1. The transitivity of the newly defined similarity relation ~ on submodules (used to form equivalence classes for the orbit count) is a load-bearing step for the central lower bound; the manuscript must supply an explicit verification that N ~ K and K ~ L imply N ~ L, directly from the definitions involving homomorphisms or the eigenring.
  2. In the proof that a non-fully-invariant maximal N is similar to at least |S| other distinct maximal submodules, the map from elements of the eigenring S to the corresponding similar submodules must be shown to be injective; without this, the inequality |Max(M)| >= 1 + |S| does not follow.
  3. The construction of the canonical one-to-one map Max(M) -> Max_r(End_R(M)) (appearing after the similarity results) should explicitly address compatibility with the similarity relation, as this map is invoked in the applications to matrix rings.
minor comments (2)
  1. Define the notation E = End_R(M) at its first appearance to prevent confusion with the eigenring S of a submodule.
  2. In the applications section, clarify whether the infinite algebras are assumed to be over a field or more general base rings when discussing matrix rings.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will incorporate the necessary clarifications and proofs in a revised version.

read point-by-point responses

  1. Referee: The transitivity of the newly defined similarity relation ~ on submodules (used to form equivalence classes for the orbit count) is a load-bearing step for the central lower bound; the manuscript must supply an explicit verification that N ~ K and K ~ L imply N ~ L, directly from the definitions involving homomorphisms or the eigenring.

    Authors: We agree that transitivity requires an explicit verification from the definitions. In the revised manuscript we will add a dedicated lemma proving that N ~ K and K ~ L imply N ~ L, using the given homomorphisms between the submodules and the corresponding eigenrings. revision: yes

  2. Referee: In the proof that a non-fully-invariant maximal N is similar to at least |S| other distinct maximal submodules, the map from elements of the eigenring S to the corresponding similar submodules must be shown to be injective; without this, the inequality |Max(M)| >= 1 + |S| does not follow.

    Authors: We acknowledge that injectivity of the indicated map must be established explicitly. We will insert a short argument showing that distinct elements of the eigenring S produce distinct similar maximal submodules, thereby justifying the bound |Max(M)| >= 1 + |S|. revision: yes

  3. Referee: The construction of the canonical one-to-one map Max(M) -> Max_r(End_R(M)) (appearing after the similarity results) should explicitly address compatibility with the similarity relation, as this map is invoked in the applications to matrix rings.

    Authors: We will revise the paragraph containing the canonical map to include a brief discussion of its compatibility with the similarity relation, noting how the map behaves on similarity classes and confirming that the applications to matrix rings remain valid under this compatibility. revision: yes

Circularity Check

0 steps flagged

No circularity; bounds derived from new similarity relation and standard module properties

full rationale

The paper introduces a similarity relation on submodules extending the classical notion for right ideals, defines the eigenring S of a maximal submodule N, and proves that non-fully-invariant N is similar to at least 1+|S| distinct maximal submodules. This counting follows from the properties established for the relation and the action of S rather than reducing by construction to a fitted input or self-referential definition. No load-bearing step matches the enumerated circular patterns; the central lower bound on |Max(M)| is obtained via independent proofs using the new definitions together with standard facts about projective modules. The derivation is self-contained against external benchmarks with no self-citation chains or ansatz smuggling required for the main claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard definitions from module theory together with the newly introduced similarity relation; no free parameters or invented entities are introduced.

axioms (1)
  • standard math Projective modules, maximal submodules, eigenrings, Artinian rings, and finite length are defined and behave as in classical ring and module theory.
    All stated results invoke these background notions without re-proving them.

pith-pipeline@v0.9.0 · 5581 in / 1189 out tokens · 117762 ms · 2026-05-16T02:24:13.894108+00:00 · methodology

discussion (0)

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Reference graph

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