Definition:Irreducible Polynomial

Definition
Let $R$ be an integral domain.

An irreducible polynomial over $R$ is an irreducible element of the polynomial ring $R[X]$.

Also see
By Units of Ring of Polynomial Forms over Field, a polynomial in a single indeterminate with coefficients in a field is irreducible it is not a product of two polynomial forms of smaller degree.

This is not necessarily true for polynomials over a commutative ring.