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arxiv: 2604.20820 · v1 · submitted 2026-04-22 · 🧮 math.AC

Recognition: unknown

On S-Prime Element Principle

Chetan Patil, Sachin Sarode, Vinayak Joshi

Pith reviewed 2026-05-09 22:22 UTC · model grok-4.3

classification 🧮 math.AC
keywords S-prime elementsV-latticesmultiplicative latticesprime elementsS-Prime Element Principle
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The pith

The S-Prime Element Principle provides a uniform way to prove existence of prime elements in multiplicative lattices by setting S to {1}.

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

The paper defines S-prime elements inside V-lattices for any multiplicatively closed subset S. It then states the S-Prime Element Principle as a criterion that directly verifies when a given element satisfies the S-prime condition. Specializing the principle to the case S equals the singleton containing only the top element recovers, in one step, many earlier theorems on the existence of prime elements in multiplicative lattices.

Core claim

We introduce S-prime elements in V-lattices, where S is a multiplicatively closed subset of a V-lattice L. We introduce the S-Prime Element Principle to prove that certain elements in V-lattices are S-prime elements. This principle leads to a direct and uniform approach to the results on the existence of prime elements in multiplicative lattices when S={1}.

What carries the argument

The S-Prime Element Principle, a criterion that certifies S-primeness of elements inside V-lattices and specializes to the ordinary prime-element case when S={1}.

If this is right

  • Separate case-by-case arguments for prime elements in multiplicative lattices can be replaced by a single invocation of the principle when S={1}.
  • The same criterion applies unchanged to any multiplicatively closed S, extending the classical theory beyond the S={1} restriction.
  • Proofs that previously required special lattice properties now follow from the V-lattice axioms plus the new principle.

Where Pith is reading between the lines

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

  • The framework may supply a template for similar uniform principles in other lattice-based algebraic structures.
  • Applications to ideal lattices of rings could follow if the V-lattice multiplication matches ideal multiplication.

Load-bearing premise

The newly defined S-Prime Element Principle holds in general V-lattices and reduces without extra restrictions to the known case of ordinary primes when S consists only of the identity.

What would settle it

A concrete V-lattice together with an element that satisfies every hypothesis of the S-Prime Element Principle yet fails to be S-prime would refute the principle.

Figures

Figures reproduced from arXiv: 2604.20820 by Chetan Patil, Sachin Sarode, Vinayak Joshi.

Figure 1
Figure 1. Figure 1: N5 In the lattice N5, the multiplication do not distribute over join. This motivates us to define a broader class of lattices. Definition 1.3. A complete lattice L is said to be a V -lattice if there exists a commutative, associative binary operation “⋅” on L such that: (1) a ≤ b Ô⇒ a ⋅ c ≤ b ⋅ c, for a, b, c ∈ L. (2) a ⋅ b ≤ a ∧ b, for a, b ∈ L. (3) a ⋅ 1 = a for every a ∈ L. Remark 1.4. Every multiplicat… view at source ↗
Figure 2
Figure 2. Figure 2: L = Id(Z12) Let F be an S-Ako or S-Oka element family of a V -lattice L, then F ′ denotes the complement of F (i.e. the set consisting of all elements of L that does not belong to F). Throughout this paper, we assume that if F is an S-Ako or an S-Oka family in a V -lattice L with 1 compact, then F′ has a maximal element in L, i.e., M ax(F′ ) /= ∅. Theorem 3.8 (S–Prime Element Principle (S–PEP)). Let L be a… view at source ↗
Figure 3
Figure 3. Figure 3: A multiplicative lattice K Theorem 4.12. Let S = {1}, a multiplicatively closed subset of a c-lattice L. Then an element maximal with respect to being annihilator element of L is a prime element of L. Proof. Let F be a set of all non annihilator elements of L. Since 1 is not an annihilator element, S ⊆ F. We claim that F is an S-Oka family. Let (i∨j), (i ∶ j) ∈ F. Therefore (i∨j) /= (0 ∶ x) and (i ∶ j) /= … view at source ↗
read the original abstract

In this paper, we introduce $S$-prime elements in $V$-lattices, where $S$ is a multiplicatively closed subset of a $V$-lattice $L$. In addition, we introduce the $S$-Prime Element Principle to prove that certain elements in $V$-lattices are $S$-prime elements. This principle leads to a direct and uniform approach to the results on the existence of prime elements in multiplicative lattices when $S=\{1\}$.

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 / 2 minor

Summary. The paper introduces S-prime elements in V-lattices, where S is a multiplicatively closed subset of the V-lattice L, along with an S-Prime Element Principle intended to establish that certain elements are S-prime. It claims this principle yields a direct and uniform approach to known results on the existence of prime elements in multiplicative lattices in the special case S = {1}.

Significance. If the S-Prime Element Principle is valid and applies without hidden restrictions, it would supply a general tool for proving primeness in lattice-theoretic settings and unify several existence results for primes in multiplicative lattices under a single framework.

minor comments (2)
  1. The abstract refers to V-lattices and S-prime elements without providing a definition, reference, or illustrative example, making it difficult to assess the scope of the claimed generalization.
  2. No indication is given of the precise statement of the S-Prime Element Principle or the hypotheses under which it holds, which are load-bearing for the uniformity claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for highlighting the potential significance of the S-Prime Element Principle. We address the uncertainty regarding its validity and applicability below.

read point-by-point responses

  1. Referee: If the S-Prime Element Principle is valid and applies without hidden restrictions, it would supply a general tool for proving primeness in lattice-theoretic settings and unify several existence results for primes in multiplicative lattices under a single framework. The recommendation is uncertain.

    Authors: The S-Prime Element Principle is stated precisely in the manuscript and proved directly from the axioms of V-lattices and the multiplicatively closed set S. All hypotheses required for the principle are listed explicitly in its formulation, with no additional or hidden restrictions. When S = {1}, the principle specializes to a uniform argument that recovers the known existence theorems for prime elements in multiplicative lattices; we illustrate this by deriving several standard results as direct corollaries. The proof and the applications are contained in the body of the paper. revision: no

Circularity Check

0 steps flagged

No significant circularity; new principle introduced independently

full rationale

The paper introduces S-prime elements and the S-Prime Element Principle as new constructs in V-lattices, then applies the principle to recover known results on prime elements for the special case S={1}. No derivation step reduces a claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation; the central claim is that the newly stated principle yields a uniform proof, which is a standard generalization technique rather than a restatement by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities can be extracted. The work appears to rest on standard lattice axioms plus the new definitions and principle whose justification is not visible.

pith-pipeline@v0.9.0 · 5365 in / 1061 out tokens · 38880 ms · 2026-05-09T22:22:42.233585+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On $S$-Noetherian Lattices

    math.AC 2026-04 unverdicted novelty 5.0

    S-Noetherian lattices generalize Noetherian rings, with ring-lattice equivalence, S-prime compactness characterization, and unique S-primary decompositions.

Reference graph

Works this paper leans on

19 extracted references · 5 canonical work pages · cited by 1 Pith paper

  1. [1]

    Alarcon, D

    F. Alarcon, D. D. Anderson and C. Jayaram,Some results on abstract commutative ideal theory, Period. Math. Hung.,30(1995), 1-26. https://doi.org/10.1007/BF01876923

    work page doi:10.1007/bf01876923 1995

  2. [2]

    D. D. Anderson,Abstract commutative ideal theory without chain condition, Algebra Univers.,6(1976), 131-145. https://doi.org/10.1007/BF02485825

    work page doi:10.1007/bf02485825 1976

  3. [3]

    R. G. Burton,Fractional elements in multiplicative lattices, Pacific J. Math.56 (1)(1975), 35-49

    1975

  4. [4]

    C˘ alug˘ areanu,Lattice concepts of module theory, (Birkh¨ auser, Basel, 1998)

    G. C˘ alug˘ areanu,Lattice concepts of module theory, (Birkh¨ auser, Basel, 1998)

    1998

  5. [5]

    I. S. Cohen,Commutative rings with restricted chain condition, Duke Math. J. 17 (1950) 27–42

    1950

  6. [6]

    R. P. Dilworth,Abstract commutative ideal theory, Pacific. J. Math.12(1962), 481-498

    1962

  7. [7]

    Hamed and A

    A. Hamed and A. Malek,S-prime ideals of a commutative ring, Beitr. Algebra Geom.,61(2020),533-542

    2020

  8. [8]

    Joshi and S

    V. Joshi and S. Ballal,A note on n-Baer multiplicative lattices, Southeast Asian Bull. Math.39(2015), 67-76

    2015

  9. [9]

    Joshi and S

    V. Joshi and S. Sarode,Beck’s conjecture and multiplicative lattices, Discrete Math.338 (3)(2015), 93-98

    2015

  10. [10]

    Jayaram,Primary elements in Pr¨ ufer lattices, Czech

    C. Jayaram,Primary elements in Pr¨ ufer lattices, Czech. Math. J.52 (127)(2002), 585-593

    2002

  11. [11]

    Jayaram and E

    C. Jayaram and E. W. Johnson,s-prime elements in multiplicative lattices, Period. Math. Hung.31(1995), 201-208

    1995

  12. [12]

    Jayaram and E

    C. Jayaram and E. W. Johnson,Strong compact elements in multiplicative lattices, Czech. Math. J.47 (122)(1997), 105-112

    1997

  13. [13]

    Jayaram, U

    C. Jayaram, U. Tekir and E. Yetkin,2-absorbing and weakly 2-absorbing elements in multiplicative lattices, Commun. Algebra,42 (6)(2014), 2338-2353. https://doi.org/10.1080/00927872.2012.761229

    work page doi:10.1080/00927872.2012.761229 2014

  14. [14]

    Kavishwar and V

    S. Kavishwar and V. Joshi,A note on prime ideal principle in lattices, Asian-European J. Math.8 (2)(2015) 1550024 (12 pages)

    2015

  15. [15]

    T. Y. Lam and M. L. Reyes,A prime ideal principle in commutative algebra, J. Alebra.319(2008), 3006-3027. Algebra, Volume 2014, Article ID 890312, 7 pages. https://doi.org/10.1155/2014/890312

    work page doi:10.1155/2014/890312 2008

  16. [16]

    C. S. Manjarekar and A. N. Chavan,z-elements andz j-elements in multiplicative lattices, Palestine J. Math.8 (1) (2019), 138-147

    2019

  17. [17]

    Sarode,ϕ−X- elements in multiplicative lattices, J

    S. Sarode,ϕ−X- elements in multiplicative lattices, J. Indian Math. Soc.,91 (1-2)(2024), 217-228

    2024

  18. [18]

    Sarode and V

    S. Sarode and V. Joshi,X-elements in multiplicative lattices-A generalization ofJ-ideals,n-ideals andr-ideals in rings, Int. Electron. J. Algebra, DOI:10.24330/ieja.1102289 (also see https://arxiv.org/abs/2101.06667)

    work page doi:10.24330/ieja.1102289

  19. [19]

    Ward and R

    M. Ward and R. P. Dilworth,Residuated lattices, Trans. Amer. Math. Soc.45(1939), 335-354. *Department of Mathematics, Shri Muktanand College, Gangapur, Dist. Chh. Sambhajinagar - 431 109, India. Email address:sarodemaths@gmail.com **School of Technology Management and Engineering, SVKM NMIMS Global University, Dhule-424 001, India. Email address:patilcs19...

    1939