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. Care should be taken to distinguish prime elements from irreducible elements, a similar concept which is the same in many rings.
An element of a commutative ring is said to be prime if it is not zero or a unit and whenever divides for some and in , then divides or divides . Equivalently, an element is prime if, and only if, the principal ideal generated by is a nonzero prime ideal.
Interest in prime elements comes from the Fundamental theorem of arithmetic, which asserts that each 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, the ring of Gaussian integers, since and 2 does not divide any factor on the right.
Connection with prime ideals
Prime elements should not be confused with irreducible elements. In an integral domain, every prime is irreducible but the converse is not true in general. However, in unique factorization domains, or more generally in GCD domains, primes and irreducibles are the same.
The following are examples of prime elements in rings:
- The integers ±2, ±3, ±5, ±7, ±11,... in the ring of integers Z
- the complex numbers (), 19, and () in the ring of Gaussian integers Z
- the polynomials and in the ring of polynomials over Z.
- Section III.3 of Hungerford, Thomas W. (1980), Algebra, Graduate Texts in Mathematics 73 (Reprint of 1974 ed.), New York: Springer-Verlag, ISBN 978-0-387-90518-1, MR 0600654
- Jacobson, Nathan (1989), Basic algebra. II (2 ed.), New York: W. H. Freeman and Company, pp. xviii+686, ISBN 0-7167-1933-9, MR 1009787