I am taking a first course in algebra and I am having issues with a detail in this proof that every finite domain is a division ring.(adsbygoogle = window.adsbygoogle || []).push({});

The argument that I used is that (because of cancellation in domains) left & right multiplication by a nonzero element r in a domain R gives a bijection from R to rR and R to Rr, respectively.

Since rR and Rr are contained in R, we must have rR = R = Rr. If I knew that 1 was in R at this point I would be done since I could argue that each element has a right and left inverse in R.

But, I don't know how to prove that 1 is in R, so all I can say is that for each pair of elements r and x, there are elements a and b such that ra = x and br = x.

Choosing x = r, for each r, there are elements such that ra = r and br = r. This seems potentially useful in showing that 1 is in R but I cannot complete the proof!

Any help at all is greatly appreciated. :)

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# Every finite domain is a division ring

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