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I want to see if there is any elementary way to prove the following assertion about matrices over division rings (such as not using Wedderburn's theory or tensoring techniques).

If an $n\times n$ matrix over a division ring has left inverse, then it also has right inverse.

The assertion has elementary proof for matrices over fields, but I am considering over division rings.

One can give some hints also.

Beginner
  • 10,836
  • It's a Noetherian ring, and in Noetherian ring one-sided inverses are two-sided. See https://math.stackexchange.com/questions/63609/invertibility-of-elements-in-a-left-noetherian-ring – Angina Seng May 09 '20 at 14:06
  • The duplicate is not explicitly asking about division rings but nevertheless all the arguments hold. An answer to this question would necessarily cause needless duplication. – rschwieb May 09 '20 at 14:17

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