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 $$D$$.