On power sums of matrices over a finite commutative ring

2015/05/29 by Fortuny, P., Grau, J. M., Oller-Marcén, A. M. +1
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1505.08132

Abstract

In this paper we deal with the problem of computing the sum of the k-th powers of all the elements of the matrix ring \mathbbM_d(R) with d>1 and R a finite commutative ring. We completely solve the problem in the case R=ℤ/nℤ and give some results that compute the value of this sum if R is an arbitrary finite commutative ring R for many values of k and d. Finally, based on computational evidence and using some technical results proved in the paper we conjecture that the sum of the k-th powers of all the elements of the matrix ring \mathbbM_d(R) is always 0 unless d=2, \textrmcard(R) ≡ 2 \pmod 4, 1

Related

• Power sums over commutative and unitary rings
• Matrices over commutative rings as sum of higher powers
• Matrices over a commutative ring as sums of three idempotents or three involutions
• Idempotents of 2× 2 matrix rings over rings of formal power series
• Commutators of finite multiplicative order