Irreducible element

In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, a non-zero non-unit

Unit (ring theory)

In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...

 element in an integral domain is said to be irreducible if it is not a product of two non-units.

Irreducible elements should not be confused with prime element

Prime element

In abstract algebra, an element p of a commutative ring R is said to be prime if it is not zero, not a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b...

s. (A non-unit element in a commutative ring

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

  is called prime if whenever for some and in , then or .) Every prime element is irreducible, but the converse is not true in general. The converse is true for UFD

Unique factorization domain

In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...

s (or, more generally, GCD domain

GCD domain

In mathematics, a GCD domain is an integral domain R with the property that any two non-zero elements have a greatest common divisor . Equivalently, any two non-zero elements of R have a least common multiple ....

s.)

Moreover, while an ideal generated by a prime element is a prime ideal

Prime ideal

In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal

Irreducible ideal

In mathematics, an ideal of a commutative ring is said to be irreducible if it cannot be written as a finite intersection of ideals properly containing it....

. However, if is a GCD domain, and is an irreducible element of , then the ideal generated by is an irreducible ideal of .

Example

In the quadratic integer ring , it can be shown using norm

Field norm

In mathematics, the norm is a mapping defined in field theory, to map elements of a larger field into a smaller one.-Formal definitions:1. Let K be a field and L a finite extension of K...

arguments that the number 3 is irreducible. However, it is not a prime in this ring since, for example,

but does not divide either of the two factors.