In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials.
Interest in prime elements comes from the fundamental theorem of arithmetic, which asserts that each nonzero integer can be written in essentially only one way as 1 or −1 multiplied by a product of positive prime numbers.
This led to the study of unique factorization domains, which generalize what was just illustrated in the integers.
Being prime is relative to which ring an element is considered to be in; for example, 2 is a prime element in Z but it is not in Z[i], the ring of Gaussian integers, since 2 = (1 + i)(1 − i) and 2 does not divide any factor on the right.
In an integral domain, every prime is irreducible[2] but the converse is not true in general.