What is the most right way to define irreducible polynomial over arbitrary ring?

Question

Let $R$ be an arbitrary ring (not necessarily commutative) with identity and let $f(x)\in R[x]$ be a polynomial over $R$.

Then we have several analogues for irreducibility of $f(x)$. For example,

1. $R[x]f(x)$ is maximal left ideal in $R[x]$;
2. $f(x)R[x]$ is maximal right ideal in $R[x]$;
3. If $f(x) = g(x)h(x)$ for non-constant polynomials $g,h\in R[x]$ then $g$ or $h$ is invertible polynomial.
4. If $f(x) = g(x)h(x)$ for non-constant polynomials $g,h\in R[x]$ then $\deg(g)\geq \deg(f)$ or $\deg(h)\geq \deg(f)$.

So, what is the most right or logical way to define iireducible polynomial over arbitrary ring? Maybe there are another good definitions?