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The concept of a unique factorization domain (UFD) has been defined, for commutative (e.g. (4) page 21) and non-commutative (1) integral domains. We take the theory a stage further here by defining a unique factorization ring (UFR), where throughout, a ring is understood to mean a commutative ring with identity, possibly containing proper zero-divisors.
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Colin R. Fletcher (1969) studied this question.
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