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 iff it has no non-trivial factorization in $D$.

This can alternatively be stated:
 * $x$ is irreducible iff the only divisors of $x$ are its associates and the units of $D$.
 * $x$ is irreducible iff it has no proper divisors.

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