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Distribution and Generalized Center in Planar Nearrings

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Tim Boykett

Rings and Algebras · Submitted 5 Jul 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim If a planar nearring has nonzero distributive elements, then it is an extension of its zero multiplier part by an abelian group.

how it is stated · asserted 1.00 read the record →

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arXiv ID1607.01204 (version 2)
Pith IDRV3SAPLM, Pith's permanent record of this paper and its review
CitationTim Boykett. "Distribution and Generalized Center in Planar Nearrings." arXiv:1607.01204 (2016). https://pith.science/paper/RV3SAPLM
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,520 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.51 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.70 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Planar nearrings play an important role in nearring theory, both from the structural side as being close to generalised nearfields, as well as from an applications perspective, in geometry and designs. We investigate the distributive elements of planar nearrings. If a planar nearring has nonzero distributive elements, then it is an extension of its zero multiplier part by an abelian group. In the case that there are distributive elements that are not zero multipliers, then this extension splits, giving an explicit description of the nearring. This generalises the structure of planar rings. We provide a family of examples where this does not occur, the distributive elements being precisely the zero multipliers. We apply this knowledge to the question of determining the generalized center of planar nearrings as well as finding new proofs of other older results.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Planar nearrings play an important role in nearring theory, both from the structural side as being close to generalised nearfields, as well as from an applications perspective, in geometry and designs. background 1.00 0.04 0.03 asserted 1.00
s1 We investigate the distributive elements of planar nearrings. contribution 1.00 0.88 0.04 asserted 0.68
s2 If a planar nearring has nonzero distributive elements, then it is an extension of its zero multiplier part by an abelian group. result 1.00 0.81 0.37 asserted 1.00
s3 In the case that there are distributive elements that are not zero multipliers, then this extension splits, giving an explicit description of the nearring. result 1.00 0.86 0.23 asserted 1.00
s4 This generalises the structure of planar rings. interpretation 0.95 0.75 0.13 asserted 1.00
s5 We provide a family of examples where this does not occur, the distributive elements being precisely the zero multipliers. result 0.49 0.93 0.08 asserted 1.00
s6 We apply this knowledge to the question of determining the generalized center of planar nearrings as well as finding new proofs of other older results. contribution 0.96 0.93 0.17 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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  • references and citing sentences
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  • T body sentences
  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Planar nearrings play an important role in nearring theory, both from the structural side as being close to generalised nearfields, as well as from an applications perspective, in geometry and designs. We investigate the distributive elements of planar nearrings. If a planar nearring has nonzero distributive elements, then it is an extension of its zero multiplier part by an abelian group. In the case that there are distributive elements that are not zero multipliers, then this extension splits, giving an explicit description of the nearring. This generalises the structure of planar rings. We provide a family of examples where this does not occur, the distributive elements being precisely the zero multipliers. We apply this knowledge to the question of determining the generalized center of planar nearrings as well as finding new proofs of other older results.

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