Prime elements in integral domains

[Math Amateur]

In Dummit and Foote, Section 8.3 on Unique Factorization Domains, Proposition 10 reads as follows:

Proposition 10: In an integral domain a prime element is always irreducible.

The proof reads as follows:

===========================================================

Suppose (p) is a non-zero prime ideal and p = ab.

Then ab = p \in (p), so by definition of prime ideal, one of a or b, say a, is in (p).

Thus a = pr for some r.

This implies p = ab = prb and so rb = 1 and b is a unit.

This shows that p is irreducible.

==============================================================

My question is as follows: Where in this proof do D&F use the fact that p is in an integral domain? (It almost reads as if this applies for any ring)

Peter

 

[Siron]

In this step:

This implies p = ab = prb and so rb = 1 and b is a unit.

Since an integral domain has no zero divisors by definition there's a cancelation law which says:
Let R be an integral domain and a,b,c \in R. If a \neq 0 and ab=ac then b=c.