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 concept that is the same in unique factorization domains but not the same in general.

Definition

An element of a commutative ring is said to be prime if it is not the zero element or a unit, and for all in , whenever divides , divides or divides (that is, <math>p\mid ab \implies p\mid a\ \or\ p\mid b </math>). With this definition, Euclid's lemma is the assertion that prime numbers are prime elements in the ring of integers. Equivalently, an element is prime if, and only if, the principal ideal generated by is a nonzero prime ideal. (In an integral domain, the ideal is a prime ideal, but is not considered to be a prime element.) Note: References defining primality for an element <math>p\in R</math> often restrict to be an integral domain or a Euclidean domain, or may add the additional requirement that is not a zero-divisor.

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 but it is not in , the ring of Gaussian integers, since and 2 does not divide any factor on the right.

Connection with prime ideals

An ideal in the ring (with unity) is prime if the factor ring is an integral domain. Equivalently, is prime if whenever <math>ab \in I</math> then either <math>a \in I</math> or <math>b \in I</math>.

A nonzero principal ideal is prime if and only if it is generated by a prime element.

Irreducible elements

Prime elements should not be confused with irreducible elements. Recall that an element of an integral domain is irreducible if it is not a unit and whenever , either or is a unit, while several non-equivalent definitions of irreducibility of varying strength exist for elements of general commutative rings (see the main article). 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.

Examples

The following are examples of prime elements in rings:

  • The integers , , , , , ... in the ring of integers
  • the complex numbers , , and in the ring of Gaussian integers
  • the polynomials and in , the ring of polynomials over .
  • 2 in the quotient ring
  • is prime but not irreducible in the ring
  • In the ring of pairs of integers, is prime but not irreducible (one has ).
  • In the ring of algebraic integers <math>\mathbf Z[\sqrt{-5}],</math> the element is irreducible but not prime (as 3 divides <math>9=(2+\sqrt{-5})(2-\sqrt{-5})</math> and 3 does not divide any factor on the right).

References

;Notes

;Sources

  • Section III.3 of