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Nilpotent Product Probability of Finite Rings

  • Dibyasman Sarma
  • Tikaram Subedi
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Abstract

For a finite ring R, we investigate the probability that the product of two randomly chosen elements in R is nilpotent, which we call the nilpotent product probability (NPP) of R and denote by Pnil(R). We derive bounds for Pnil(R) that depend on the structure of R, and prove that among all commutative rings with identity, the class of local rings with residue field ℤ2 attains the maximum value of NPP, which is equal to 3/4. Further, we determine the set of all values of NPP for commutative rings with identity and classify all rings attaining those values. Finally, we investigate the relationship between NPP of R and NPP of some subrings of the matrix ring over R.

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Cite

BibTeXRIS
BibTeX
@misc{sarma2026nilpotent,
  title = {{Nilpotent Product Probability of Finite Rings}},
  author = {Dibyasman Sarma and Tikaram Subedi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07892},
  eprint = {2610.07892},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07892},
}
RIS
TY  - UNPB
TI  - Nilpotent Product Probability of Finite Rings
AU  - Dibyasman Sarma
AU  - Tikaram Subedi
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.07892
UR  - https://arxiv.org/abs/2610.07892
AB  - For a finite ring R, we investigate the probability that the product of two randomly chosen elements in R is nilpotent, which we call the nilpotent product probability (NPP) of R and denote by P_nil(R). We derive bounds for P_nil(R) that depend on the structure of R, and prove that among all commutative rings with identity, the class of local rings with residue field ℤ_2 attains the maximum value of NPP, which is equal to 3/4. Further, we determine the set of all values of NPP for commutative rings with identity and classify all rings attaining those values. Finally, we investigate the relationship between NPP of R and NPP of some subrings of the matrix ring over R.
ER  -