PreprintCC BY 4.0arXiv
Matrices over non-commutative rings as sums of fifth and seventh powers
Abstract
Let R be non-commutative ring with unity. In this paper, we prove that a given matrix in Mn(R) is a sum of p-th powers in Mn(R), where n < p if and only if its trace can be written as a sum of p-th powers in Mn(R), for n = 2,3,4 and p = 5 and n = 4,5,6 for p = 7. This paper extends the results of S. A. Katre and Kshipra Wadikar, (p=3) and Garge (n ≥ p), (p = 5, 7), and S. A. Katre and Deepa Krishnamurthi (n ≥ p).
Related
- q-Supercongruences Modulo the Third and Fifth Powers of a Cyclotomic Polynomial
- Commutative coherent rings with 𝒢𝒫not⊆𝒢ℱ
- Finite Non-Commutative Centrally Essential Rings of Minimal Order
- Commutative rings with square-difference factor absorbing factorization
- Contravariantly finite resolving subcategories over commutative local rings are trivial ones
BibTeX
@misc{kochrekar2026matrices,
title = {{Matrices over non-commutative rings as sums of fifth and seventh powers}},
author = {Tanay Kochrekar and A. S. Garge},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14064},
eprint = {2609.14064},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14064},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Matrices over non-commutative rings as sums of fifth and seventh powers
AU - Tanay Kochrekar
AU - A. S. Garge
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14064
UR - https://arxiv.org/abs/2609.14064
AB - Let R be non-commutative ring with unity. In this paper, we prove that a given matrix in M_n(R) is a sum of p-th powers in M_n(R), where n < p if and only if its trace can be written as a sum of p-th powers in M_n(R), for n = 2,3,4 and p = 5 and n = 4,5,6 for p = 7. This paper extends the results of S. A. Katre and Kshipra Wadikar, (p=3) and Garge (n ≥ p), (p = 5, 7), and S. A. Katre and Deepa Krishnamurthi (n ≥ p).
ER -