An element p
of a commutative integral domain R
if it is nonzero and whenever p|ab
Here we write x|y for x divides y.
For example, prime numbers are prime elements of the
ring of integers.
Lemma A prime element of R is irreducible
Suppose that p is prime but that it is not irreducible. Then there
exists non-units x,y in R such that p=xy. Clearly
then p|xy. Since p is prime it divides one of x
or y. Without loss of generality, it divides x. Thus
pa=x, for some a in R. Thus we have p=pay
or p(1-ay)=0. Since R is an integral domain it follows
that y must be a unit. This contradicts the choice of y
and proves the lemma.