In rings with zero-divisors, factorization theory is much more complicated than in domains, e.g. $\rm\:x = (3+2x)(2-3x)\in \Bbb Z_6[x].\:$ Basic notions such as associate and irreducible bifurcate into at least a few inequivalent notions, e.g. see the papers below, where three different notions of associateness are compared:
- $\ a\sim b\ $ are $ $ associates $ $ if $\, a\mid b\,$ and $\,b\mid a$
- $\ a\approx b\ $ are $ $ strong associates $ $ if $\, a = ub\,$ for some unit $\,u.$
- $\ a \cong b\ $ are $ $ very strong associates $ $ if $\,a\sim b\,$ and $\,a\ne 0,\ a = rb\,\Rightarrow\, r\,$ unit
When are Associates Unit Multiples?
D.D. Anderson, M. Axtell, S.J. Forman, and Joe Stickles.
Rocky Mountain J. Math. Volume 34, Number 3 (2004), 811-828.
Factorization in Commutative Rings with Zero-divisors.
D.D. Anderson, Silvia Valdes-Leon.
Rocky Mountain J. Math. Volume 28, Number 2 (1996), 439-480