Definition:Irreducible Element of Ring

Definition
Let $\left({D, +, \circ}\right)$ be an integral domain whose zero is $0_D$.

Let $\left({U_D, \circ}\right)$ be the group of units of $\left({D, +, \circ}\right)$.

Let $x \in D: x \notin U_D, x \ne 0_D$, that is, $x$ is non-zero and not a unit.

Then $x$ is defined as irreducible it has no non-trivial factorization in $D$.

The definition can alternatively be stated:
 * $x$ is irreducible the only divisors of $x$ are its associates and the units of $D$.
 * $x$ is irreducible it has no proper divisors.
 * $x$ is irreducible it cannot be written as a product of two non-units.

Also defined as
Some sources define the concept of irreducibility only when an integral domain $\left({D, +, \circ}\right)$ is Euclidean.

Also known as
An irreducible element of a ring is also known as an atom of a ring, hence described as atomic.

Also see

 * Definition:Irreducible Ring
 * Definition:Prime Element of Ring

Special cases

 * Definition:Irreducible Polynomial