An element

*p* of a

commutative integral domain *R* is called

prime if it is nonzero and whenever

*p|ab* in

*R* then
either

*p|a* or

*p|b*.

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

*Proof*
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.