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arxiv: 1907.06047 · v1 · pith:OUAIHNNTnew · submitted 2019-07-13 · 🧮 math.RA

Semimodules over commutative semirings and modules over unitary commutative rings

Ivan Chajda , Helmut L\"anger This is my paper

Pith reviewed 2026-05-24 22:06 UTC · model grok-4.3

classification 🧮 math.RA
keywords semimodulessubsemimodulesclosed subsemimodulessplitting subsemimoduleslatticesposetsprojectionscommutative semirings
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The pith

Splitting submodules of modules over unitary commutative rings correspond bijectively to projections.

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

The paper examines closed and splitting subsemimodules within semimodules over commutative semirings, along with the parallel notions for modules over unitary commutative rings. It establishes that the full collection of subsemimodules forms a lattice under inclusion, the closed subsemimodules likewise form a lattice, and the splitting subsemimodules form a poset. In the module case the authors prove a natural bijection between the poset of splitting submodules and the poset of projections. A reader would care because the constructions supply explicit order-theoretic descriptions of subobjects that extend classical ideas from module theory to the semiring setting.

Core claim

We study the so-called closed and splitting subsemimodules and submodules of a given semimodule or module, respectively. We describe lattices of subsemimodules and of closed subsemimodules and posets of splitting subsemimodules and submodules. In the case of modules a natural bijective correspondence between these posets and posets of projections is established.

What carries the argument

Closed subsemimodules and splitting subsemimodules, which determine the lattice and poset structures, together with the bijection to projections in the module setting.

If this is right

  • The set of all subsemimodules of a semimodule forms a lattice under inclusion.
  • The subset of closed subsemimodules likewise forms a lattice.
  • The splitting subsemimodules form a poset under inclusion.
  • For modules the poset of splitting submodules stands in natural bijection with the poset of projections.

Where Pith is reading between the lines

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

  • The lattice and poset descriptions may permit direct transfer of fixed-point or closure arguments from semimodule theory to module theory and back.
  • The explicit bijection supplies a dictionary that could convert questions about splitting submodules into equivalent questions about idempotent endomorphisms.
  • The constructions remain available whenever the base structures are commutative and unital, so the same order-theoretic tools apply uniformly across the semiring-to-ring spectrum.

Load-bearing premise

The notions of closed and splitting subsemimodules and submodules are well-defined once the underlying semirings and rings are commutative and satisfy the unitarity conditions in the title.

What would settle it

A concrete unitary commutative ring together with a module for which the poset of splitting submodules fails to be in bijection with the poset of projections would refute the claimed correspondence.

read the original abstract

We study the so-called closed and splitting subsemimodules and submodules of a given semimodule or module, respectively. We describe lattices of subsemimodules and of closed subsemimodules and posets of splitting subsemimodules and submodules. In the case of modules a natural bijective correspondence between these posets and posets of projections is established.

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

0 major / 3 minor

Summary. The manuscript examines closed and splitting subsemimodules of semimodules over commutative semirings, together with the corresponding notions for modules over unitary commutative rings. It describes the lattices of all subsemimodules and of closed subsemimodules, as well as the posets of splitting subsemimodules and submodules. For modules it establishes a natural bijective correspondence between the poset of splitting submodules and the poset of projections.

Significance. The lattice and poset descriptions for semimodules extend classical module-theoretic constructions to a setting where addition is not necessarily cancellative. The bijection recovered for modules is the standard identification of direct summands with images of idempotent endomorphisms; the semimodule results, if correctly formulated, supply the analogous framework under the stated commutativity and unitarity hypotheses. The work therefore supplies a uniform language that may be applied in areas such as tropical linear algebra.

minor comments (3)
  1. [Introduction] The abstract states that lattices of subsemimodules and closed subsemimodules are described, yet the introduction does not indicate whether these lattices are shown to be distributive, modular, or to satisfy any other standard lattice-theoretic property; a brief statement of the main structural results would help the reader.
  2. [Section 2] Notation for the endomorphism semiring (or ring) of a semimodule (module) is introduced without an explicit symbol; subsequent references to projections would be clearer if a consistent symbol such as End(S) or End_R(M) were fixed early.
  3. [Definition 3.4] The definition of a splitting subsemimodule is given in terms of a direct sum decomposition, but the text does not explicitly verify that the complement is again a subsemimodule under the commutativity assumption; a short verification paragraph would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its potential applications, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper describes lattices and posets of closed/splitting subsemimodules and submodules, then establishes a natural bijective correspondence (for modules) between the poset of splitting submodules and the poset of projections. This matches the standard module-theoretic identification of direct summands with images of idempotent endomorphisms in End_R(M), which holds precisely under the commutativity and unitarity conditions stated in the title. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or claim; the result is a direct algebraic correspondence without reduction to prior fitted data or author-specific uniqueness theorems. The derivation is therefore self-contained against external benchmarks in ring and module theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no details on any free parameters, axioms, or invented entities used in the work.

pith-pipeline@v0.9.0 · 5579 in / 1059 out tokens · 24537 ms · 2026-05-24T22:06:10.958983+00:00 · methodology

discussion (0)

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Reference graph

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