Null polynomials over a finite ring need not form a two-sided ideal

Valentin Havlovec

arXiv:2608.13251v1 Primary: math.RA Secondary: math.AC

Abstract

We give an example of a finite ring for which the set of polynomials inducing the zero function fails to be a two-sided ideal, disproving a conjecture of Werner.

Disclosure

“btain the module-finite, free Z-algebra S. Then R ∼ = S/2S. By [4, Theorem 2.4], the failure of N(R) to be a two-sided ideal implies that the integer-valued polynomials over S do not form a ring. Declaration on the use of LLMs The counterexample and an initial proof were generated by GPT-5.6 Sol. The author subsequently verified the construction independently and simplified the proof. The author takes full responsibility for the mathematical content of the art”
PDF page 4

Classification: Substantial mathematical content or result generation
Multiplier: 10
Verified: 2026-08-23T05:13:07.110228Z

Structural counts

Pages 4 pdf

Theorems 1 source

Lemmas 0 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 17 source

Bibliography entries 16 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file nullpolscounterex.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.