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arxiv: 2606.11932 · v1 · pith:J74DTMB6new · submitted 2026-06-10 · 🧮 math.AC

On S-prime and S-primary elements in multiplicative lattices

Sachin Sarode , Chetan Patil , Vinayak Joshi This is my paper

Pith reviewed 2026-06-27 07:50 UTC · model grok-4.3

classification 🧮 math.AC
keywords multiplicative latticesS-prime elementsweakly S-prime elementsS-primary elementsideal latticecommutative ringsweakly primary elements
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The pith

Weakly S-prime ideals of a ring R correspond exactly to weakly S_L-prime elements in the ideal lattice Id(R).

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

The paper extends the study of S-prime and S-primary elements from rings to the setting of multiplicative lattices and introduces weakly S-prime and weakly S-primary elements as further generalizations. It proves a precise bijection: the weakly S-prime ideals of any commutative ring R with 1 stand in one-to-one correspondence with the weakly S_L-prime elements of Id(R), where S_L consists of the principal ideals generated by elements of S; the same holds for the primary versions. A sympathetic reader cares because the result lets lattice-theoretic statements about these elements translate directly into statements about ideals in ordinary rings.

Core claim

The paper claims that the weakly S-prime ideals of a commutative ring R with identity correspond precisely to the weakly S_L-prime elements of the multiplicative lattice Id(R), and likewise the weakly S-primary ideals correspond to the weakly S-primary elements, where S_L = {(s) | s ∈ S}.

What carries the argument

The multiplicative lattice Id(R) of all ideals of R under ideal multiplication, together with the map sending each ideal I of R to the element (I) in Id(R) and the definitions of weakly S_L-prime and weakly S-primary elements in that lattice.

Load-bearing premise

The ideal lattice Id(R) with ideal multiplication is a multiplicative lattice in which the definitions of weakly S_L-prime and weakly S-primary elements are well-defined and the correspondence map preserves the relevant properties.

What would settle it

A concrete commutative ring R with 1, a multiplicative set S, and an ideal I such that I is weakly S-prime in R but the corresponding element of Id(R) fails to be weakly S_L-prime (or vice versa).

read the original abstract

In this paper, we study $S$-prime elements and $S$-primary elements within the framework of multiplicative lattices. Furthermore, we define and explore weakly $S$-prime elements and weakly $S$-primary elements, which generalize weakly prime elements and weakly primary elements in multiplicative lattices respectively. We show that the weakly $S$-prime ideals (weakly $S$-primary ideals) of a commutative ring $R$ with $1$ correspond precisely to the weakly $S_L$-prime elements (weakly $S$-primary elements) of the ideal lattice $Id(R)$ of $R$, where $S_L = \{(s) \mid s \in S\}$.

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 S-prime and S-primary elements in multiplicative lattices, then introduces weakly S-prime and weakly S-primary elements as generalizations of the corresponding notions for multiplicative lattices. It proves that the weakly S-prime ideals (resp. weakly S-primary ideals) of a commutative unital ring R are in bijection with the weakly S_L-prime elements (resp. weakly S-primary elements) of the ideal lattice Id(R), where S_L = {(s) | s ∈ S}.

Significance. The central correspondence is obtained by direct specialization of the lattice definitions to the ideal lattice under ideal multiplication, so the result holds by construction once the notions are stated. This supplies a systematic way to transport weakly-prime-type results between commutative rings and multiplicative lattices without additional hypotheses on boundedness or distributivity.

minor comments (3)
  1. [Abstract] The abstract states the correspondence but does not record the precise lattice-theoretic definitions of weakly S-prime and weakly S-primary elements; a one-sentence indication of the definition (e.g., the condition on a·b ≤ p with a ≰ p implying b^n ≤ p or similar) would make the claim immediately verifiable from the abstract.
  2. [Preliminaries] §2 (or wherever the multiplicative-lattice axioms are recalled): the paper should cite a standard reference (e.g., Anderson–Johnson or the original Dilworth paper) for the definition of multiplicative lattice so that readers can confirm that Id(R) satisfies the required axioms.
  3. [Main theorem] The notation S_L is introduced without an explicit sentence stating that it is the image of S under the principal-ideal map; adding this sentence in the paragraph containing the correspondence theorem would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary correctly reflects the definitions and the correspondence result we establish between weakly S-prime (resp. weakly S-primary) ideals and the corresponding elements in the ideal lattice.

Circularity Check

0 steps flagged

No significant circularity; direct definitional correspondence

full rationale

The central claim is a direct correspondence between weakly S-prime (resp. S-primary) ideals of R and weakly S_L-prime (resp. S-primary) elements of Id(R). This holds by construction once the lattice multiplication is taken to be ideal product and S_L is defined as the set of principal ideals generated by S; the paper's definitions are explicitly set up to recover the ring notions under this specialization. No load-bearing self-citation, fitted parameter renamed as prediction, or self-definitional reduction appears in the abstract or described derivation. The result is a straightforward translation between two structures whose axioms are satisfied independently by Id(R).

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is populated from standard background assumptions in multiplicative lattice theory and ideal lattices of rings; no free parameters or invented entities are visible.

axioms (2)
  • standard math A multiplicative lattice is a lattice equipped with an associative, commutative multiplication that distributes over arbitrary joins.
    This is the ambient structure in which S-prime elements are defined.
  • domain assumption The set of ideals Id(R) of a commutative ring R forms a multiplicative lattice under ideal multiplication.
    Required for the correspondence to map into the lattice setting.

pith-pipeline@v0.9.1-grok · 5644 in / 1407 out tokens · 25207 ms · 2026-06-27T07:50:45.924772+00:00 · methodology

discussion (0)

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Reference graph

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