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Deciding if a ring $R$ has unique factorization or not is only considered if the ring $R$ has no zero-divisors.

The question is why ?

USUALLY we consider factorization up to units. If - for instance - we consider factorization up to units and zero-divisors we can extend the concept of Unique factorization to rings with zero-divisors.

Also we could consider Unique factorization of group rings ( with or without zero-divisors ).

mick
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    Because factorization theory is much more complicated in non-domains, e.g. basic notions like associate, irreducible, etc. bifurcate into many inequivalent notions. See this answer for some introductory expositions. – Bill Dubuque Aug 20 '16 at 19:41
  • I understand it is more complicated, but that is not a good reason imho. But thanks for the intresting reply. +1 – mick Aug 20 '16 at 20:12

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