See the paper Left Eucclidean Rings by H H Brungs, Pacific J Math 1973 <https://projecteuclid.org/download/pdf_1/euclid.pjm/1102947704> In particular on page 31 you will find the Corollary: COROLLARY. If R is a division ring then R_n is left (and right) Euclidean with φ(M) = n-rank (M) + 1 if M <> 0 and φ(0) = 0. Here R_n is the ring of n by n matrices over R. Theorem 1 is a more general statement. On Fri, Nov 23, 2018 at 3:48 PM Henry Baker <hbaker1@pipeline.com> wrote:

Euclidean algorithm / continued fraction on *square matrices* ??

3 issues: left/right variants; how to stop; how to get "smaller" remainder.

Left/right variants: I accept that non-commutativity will produce 2 variants.

How to stop: det(remainder)=0

What is the definition of "smaller" in:

remainder = dividend - quotient*divisor,

such that norm(remainder)<norm(divisor)

I.e., what should be the definition of norm(), or are there multiple definitions that work?

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