If you want to learn about localisation in general non-commutative rings then, as I understand it, the monograph cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521317139 includes more or less where people got to in the 1970s and 80s before progress halted. A little more is known if you add extra conditions.

Even more classic: no-one seems to have mentioned the insolubility of the quintic by radicals follows from the simplicity of the group A_5. Whether that is straightforward depends on what you already know.

You're right. I realised this when writing my answer and deleted it. Then I put it back again having forgotten that's why I deleted it. The colimt example is correct though. Of course it is all rather irrelevant for this question now as the correct answer is above.

Perhaps I should add that if you pass to a subgroup of index n each class can split into at most n new classes and n can occur as illustrated in my answer. I suppose an orbit-stabiliser type argument will show the number of classes must divide n.

Hmmmn. My first go at LaTeX on this seems to have failed.

Do you know examples of left Noetherian rings? There are many. I can give some if you would like.

I realised on the way home that my brain failed to move the statement of Goldie's theorem from its back to its front correctly. It only applies to semiprime (left) Goldie rings. i.e. there should be no nilpotent ideals.

I just did the check I said I didn't have time for, and yes, the non-zero divisors in a left Goldie ring (and therefore any left Noetherian ring) will be a left-reversible cancellative semigroup