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arxiv: 1907.07162 · v1 · pith:AJTWRJ6Gnew · submitted 2019-07-16 · 🧮 math.RA · math.AC

Dedekind semidomains

Peyman Nasehpour This is my paper

Pith reviewed 2026-05-24 20:37 UTC · model grok-4.3

classification 🧮 math.RA math.AC
keywords Dedekind semidomainsemiringfractional idealinvertible idealmultiplication semiringπ-semiringNoetherian semidomainsubtractive semiring
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The pith

Dedekind semodomains are semirings where every nonzero fractional ideal is invertible, equivalent to being multiplication when Noetherian.

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

The paper extends the classical Dedekind domain concept to semidomains by defining a Dedekind semidomain as a semiring in which every nonzero fractional ideal is invertible. It proves that this property holds for a Noetherian semidomain exactly when the semidomain is multiplication. Under the further subtractive condition, Dedekind semodomains are characterized as π-semirings in which every nonzero prime ideal is invertible. The work also establishes that every ideal in a subtractive Dedekind semidomain can be generated by at most two elements. These equivalences matter to readers interested in how ideal invertibility organizes algebraic structure beyond ordinary rings.

Core claim

We define Dedekind semodomains as semirings in which each nonzero fractional ideal is invertible. We prove that a Noetherian semidomain is Dedekind if and only if it is multiplication. We show that a subtractive Noetherian semidomain is Dedekind if and only if it is a π-semiring and each of its nonzero prime ideals is invertible. We also show that the maximum number of the generators of each ideal of a subtractive Dedekind semidomain is 2.

What carries the argument

Invertibility of nonzero fractional ideals, which serves as the definition of Dedekind semodomains and supplies the equivalence to multiplication and π-semiring conditions under Noetherian and subtractive hypotheses.

If this is right

  • Every ideal of a subtractive Dedekind semidomain is generated by at most two elements.
  • A Noetherian multiplication semidomain has every nonzero fractional ideal invertible.
  • A subtractive Noetherian π-semiring with every nonzero prime ideal invertible is Dedekind.
  • The two-generator bound applies uniformly to all ideals once the subtractive Dedekind condition holds.

Where Pith is reading between the lines

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

  • The two-generator result may simplify explicit calculations of ideal arithmetic in concrete semiring examples.
  • The characterizations could be tested in specific families of semirings arising from tropical or idempotent algebra.
  • If the equivalences hold, they supply a route to classify semidomains that inherit ideal-theoretic features from Dedekind domains.

Load-bearing premise

The standard definitions of semidomain, fractional ideal, invertibility, subtractive semiring, multiplication semiring, and π-semiring transfer to the semiring setting without hidden counterexamples.

What would settle it

An explicit Noetherian semidomain that is multiplication yet has at least one nonzero fractional ideal that fails to be invertible would disprove the main equivalence.

read the original abstract

We define Dedekind semidomains as semirings in which each nonzero fractional ideal is invertible. Then we find some equivalent condition for semirings to being Dedekind. For example, we prove that a Noetherian semidomain is Dedekind if and only if it is multiplication. Then we show that a subtractive Noetherian semidomain is Dedekind if and only if it is a $\pi$-semiring and each of it nonzero prime ideal is invertible. We also show that the maximum number of the generators of each ideal of a subtractive Dedekind semidomain is 2.

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 paper defines Dedekind semidomains as semirings in which every nonzero fractional ideal is invertible. It proves that a Noetherian semidomain is Dedekind if and only if it is a multiplication semiring. For subtractive Noetherian semidomains, it is Dedekind if and only if it is a π-semiring and every nonzero prime ideal is invertible. It further shows that every ideal of a subtractive Dedekind semidomain is generated by at most two elements.

Significance. If the equivalences hold, the work supplies concrete characterizations of Dedekind semidomains that parallel the classical ring-theoretic results (multiplication property, invertibility of primes, generator bounds) while incorporating the necessary adjustments for semirings. The explicit handling of the subtractive and π-semiring cases addresses the main structural distinctions from rings and yields a usable structural theorem (two-generator bound) that may be applied in semiring ideal theory.

minor comments (3)
  1. [Abstract] Abstract: the sentence 'each of it nonzero prime ideal is invertible' contains a grammatical error ('it' should be 'its').
  2. [Abstract] Abstract: the phrasing 'the maximum number of the generators of each ideal' is awkward; rephrase to 'the maximum number of generators of each ideal'.
  3. The introduction should explicitly recall or cite the definitions of 'multiplication semiring', 'π-semiring', and 'subtractive semiring' (or state that they are taken from the cited literature) so that the equivalences are immediately readable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we provide no point-by-point responses below.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces the definition of a Dedekind semidomain explicitly as a semiring in which every nonzero fractional ideal is invertible, then derives equivalent characterizations (Noetherian semidomain is Dedekind iff multiplication; subtractive Noetherian case requires π-semiring plus invertible primes; generator bound of 2) by applying standard operations on ideals and fractional ideals. These steps rely on the assumed behavior of semiring ideal theory from prior literature without any reduction of a 'prediction' or central claim back to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The equivalences are proved in both directions from the given definitions, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the new definition plus standard background from semiring and ideal theory; no free parameters or invented physical entities appear.

axioms (1)
  • standard math Standard axioms and definitions of semirings, fractional ideals, and invertibility from commutative algebra literature.
    The paper invokes these to define Dedekind semidomains and prove equivalences.
invented entities (1)
  • Dedekind semidomain no independent evidence
    purpose: Class of semirings with invertible nonzero fractional ideals
    Newly introduced definition; no independent evidence outside the paper.

pith-pipeline@v0.9.0 · 5620 in / 1345 out tokens · 24390 ms · 2026-05-24T20:37:23.780965+00:00 · methodology

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Works this paper leans on

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