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:
- the case where $R$ can be expressed as a quotient of a polynomial ring
- the case where $R$ has idempotent elements