Definition:Irreducible Element of Ring/Definition 1

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

Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$.

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

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

That is, $x$ cannot be written as a product of two non-units.

Also see

 * Equivalence of Definitions of Irreducible Element of Ring