Definition:Irreducible Element of Ring/Definition 2

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 the only divisors of $x$ are its associates and the units of $D$.

That is, $x$ has no proper divisors.

Also see

 * Equivalence of Definitions of Irreducible Element of Ring