I am searching all over and it keeps coming up how trivial it is, but I don't actually see the proof of a field being IBN. I would also like to know where a proof of a nonzero commutative ring is IBN can be found, because I keep finding in books that it is simply used as an example. I'm fine with receiving some bibliography.
2026-04-30 07:45:36.1777535136
Prove that a field/commutative ring is IBN (invariant basis number).
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As for a field, just note that modules over a field are vector spaces so this is a standard theorem in Linear Algebra.
For general commutative rings see for example exercise 11, chapter 2 in Atiyah-Macdonald's "Introduction to commutative algebra" (here it is used tensor product, but the result can be proved also without using it).