I am reading Dummit and Foote Chapter 10 - Introduction to Rings
On page 228 Corollary 3 reads:
Any finite integral domain is a field
Proof:
Let R be a finite integral domain and let a be a non-zero element of R.
By the cancellation law the map x ax is an injective function. Since R is finite, this map is surjective. In particular, there is some b such that ab = 1, i.e. a is a unit in R. Since a was an arbitrary element, R is a field.
My problem is as follows: how do we explicitly demonstrate that since R is finite, the map x ax is injective?
Can anyone please help?
Peter