Defining irreducible polynomials over polynomial rings Let R be a ring, and R[x] be a polynomial ring.
Can we define what it means for a polynomial $p(x) \in R[x]$ to be irreducible over R[x]?
Various sources (such as Wikipedia) only provide such definition for a field or a unique factorization domain. 
 A: It is not clear what irreducible means in rings that are not domains.
You can define irreducible but it won't have all properties you expect. See this.
In particular, if $R$ is not a domain, it may happen that decomposing a polynomial in $R[x]$ does not simplify it (that is, does not reduce its degree).
For instance:
$$
5x+1=(2x+1)(3x+1) \bmod 6
$$
It is even possible to decompose a linear polynomial as a product of two quadratic polynomials:
$$
x+1=(2x^2+x+7)(4x^2+6x+7) \bmod 8
$$
A: When I taught abstract algebra, I pointed out that any commutative ring with multiplicative unit could be split up into disjoint sets:
\begin{align}
1.&\quad\{0\}\\
2.&\quad\text{nonzero zero-divisors (including nilpotent elements)}\\
3.&\quad\text{units}\\
4.&\quad\text{decomposable elements: those writable as}\\
&\quad\text{product two elements not in classes 1, 2, 3}\\
5.&\quad\text{everything else.}
\end{align}
The indecomposable elements, also called irreducibles, are those in class 5.
So, for $a$ to be irreducible would mean that $a$ was not writable as product of two non-unit non-zero-divisors, and my “irreducibles” would be the “strong irreducibles” in the categorization mentioned in the link offered by @lhf.
