I was reading Gallian's Contemporary Abstract Algebra where I came across the following :
Let $D$ be an integral domain. A polynomial $f(x)\in D[x]$ that is neither the zero polynomial nor a unit is said to be irreducible over $D$ if whenever $f(x)$ is expressed as a product $f(x)=g(x)h(x)$, then $g(x)$ or $h(x)$ is a unit.
In the case that an integral domain is a field $F$, it is equivalent to define a nonconstant $f(x)\in F[x]$ to be irreducible if $f(x)$ cannot be expressed as a product of two polynomials of lower degree.
Two things changed when the integral domain is a field :
He used the term "nonconstant" instead of "neither zero nor unit".
He replaced "$f(x)=g(x)h(x)\implies g(x)$ or $h(x)$ is a unit" by "$f(x)$ cannot be expressed as a product of two polynomials of lower degree".
My questions regarding 1 and 2:
Aren't the nonzero non-invertible elements of $D[x]$ all the nonconstant polynomials anyway? I know that $2x+1$ in $\Bbb Z_4[x]$ is a non constant invertible polynomial, but $\Bbb Z_4$ isn't an integral domain. Is there such an example?
Again, could someone give an example showing that the two statements aren't the same in any $D[x]$?