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.