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 $\displaystyle \mapsto $ ax is an injective function.In particular, there is some b $\displaystyle \in $ such that ab = 1, i.e. a is a unit in R. Since a was an arbitrary element, R is a field.Since R is finite, this map is surjective.

My problem is as follows: how do we explicitly demonstrate that since R is finite, the map x $\displaystyle \mapsto $ ax is injective?

Can anyone please help?

Peter