Matrices over Non-Commutative Rings as Sums of Fifth Powers

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Sagar N. Sankeshwari, Ranjana H. Gothankar

Abstract

Let R be non-commutative ring with unity and n ≥ p ≥2, p prime. S. A. Katre, Deepa Krishnamurthi proved that an n ×n matrix over R is the sum of pth powers if and only if its trace can be written as a sum of pth powers and commutators modulo pR. This extends the results of L. N.Vaserstein (p = 2) and S. A. Katre, Kshipra Wadikar (p = 3). Also S. A. Katre, Deepa Krishnamurthi obtained necessary and sufficient conditions for a matrix over R to be written as a sum of fourth powers when n ≥2. In this paper, we obtain necessary and sufficient conditions for a matrix over R to be written as a sum of fifth powers when n ≥3.

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