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.

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: $$ x+1=(2x^2+x+7)(4x^2+6x+7) \bmod 8 $$

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.