Why does the definition of an irreducible element require us to be in an integral domain?
Why can we not define an irreducible element exactly the same in a commutative ring that is not an integral domain?
We have that an element is irreducible if it cannot be written as a product of two non-unit elements. Unit elements are well defined and unique in a commutative ring that is not an integral domain, so I cannot see that being the problem.
I've proven a proposition of my own design (probably well known and elementary, an definitely trivial). I used irreducible elements, but otherwise nothing that requires me to move from a commutative ring to an integral domain. Do irreducible elements really require me to be in an integral domain?