Turkish Journal of Mathematics
Author ORCID Identifier
ADEL ABYZOV: 0000-0002-9809-2091
STEPHEN COHEN: 0000-0002-3782-1617
PETER DANCHEV: 0000-0002-2016-2336
DANIEL TAPKIN: 0000-0003-0828-4397
DOI
10.55730/1300-0098.3543
Abstract
We significantly strengthen results on the structure of matrix rings over finite fields and applythem to describe the structure of the so-called weakly n-torsion clean rings. Specifically, we establish that, forany field F with either exactly seven or strictly more than nine elements, each matrix over F is presentableas a sum of of a tripotent matrix and a q-potent matrix if and only if each element in F is presentable as asum of a tripotent and a q-potent, whenever q > 1 is an odd integer. In addition, if Q is a power of an oddprime and F is a field of odd characteristic, having cardinality strictly greater than 9, then, for all n ≥ 1,the matrix ring Mn(F) is weakly (Q − 1)-torsion clean if and only if F is a finite field of cardinality Q.A novel contribution to the ring-theoretical theme of this study is the classification of finite fields FQof odd order in which every element is the sum of a tripotent and a potent. In this regard, we obtain anexpression for the number of consecutive triples γ − 1, γ, γ + 1 of non-square elements in FQ; in particular,FQ contains three consecutive non-square elements whenever FQ contains more than 9 elements.
Keywords
(Weakly) n-torsion clean rings, idempotents, tripotents, potents, units, finite fields, Gauss and Jacobi sums
First Page
817
Last Page
839
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
ABYZOV, ADEL; COHEN, STEPHEN; DANCHEV, PETER; and TAPKIN, DANIEL
(2024)
"Rings and finite fields whose elements are sums or differences of tripotents and potents,"
Turkish Journal of Mathematics: Vol. 48:
No.
5, Article 2.
https://doi.org/10.55730/1300-0098.3543
Available at:
https://journals.tubitak.gov.tr/math/vol48/iss5/2
