The standard definition of an irreducible element is that an element of an integral domain $D$ is irreducible if to can not be written as the factor of two non-unit elements of the ring.
However, I see no reason why this concept has to be limited to integral domains. Of course, I am sure the theory is much nicer in integral domains, but is this same notion, or a similar notion studied in non-integral domains?
For example, consider the ring $S = \mathbb{R}[j]/\langle j^2 - 1 \rangle$ (in other words, the hyperbolic numbers). This ring has zero divisors, since $(j+1)(j-1) = 0$, and hence is not an integral domain, but I would still like to study some of the properties of factorizations in $S[x]$, even if we do not have unique factorization due to the presence of zero divisors.
Has irreducibility been studied in this concept before? If so, is the same notion that is used in integral domains used, or does the concept have to be generalized?
