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Suppose we're working in a commutative ring $R$ that has zero divisors, and we have a product of elements $x = ab$.

Since $R$ has zero divisors it can't be a UFD (since every UFD is an integral domain). Hence, this opens up the possibility that $x$ admits some alternate factorization $x = cd$.

Is there some general procedure to determine when such an alternate factorization exists for a particular, or to find out what it is?

If not, then I'd be happy to get an answer for at least the following special cases:

  1. the case where $R$ can be expressed as a quotient of a polynomial ring
  2. the case where $R$ has idempotent elements
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  • $\begingroup$ An extensive theory of factorization exists for the commutative rings $\mathbb{Z}/n$. Also the Hurwitz quaternions. There are probably others, but I do not have any great knowledge of them. $\endgroup$ – SZN May 2 '17 at 5:06
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    $\begingroup$ If you work in a dedekind domain, then you can tell if the ring is not a UFD by the existence of nontrivial elements in the class group. If you study dedekind domains of finite type over an algebraically closed field then you can extend this method to products; see here, the second answer for example. $\endgroup$ – Eoin May 2 '17 at 5:33

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