摘要 : In recent decades, mathematicians have extended the definition of a unique factorization domain to rings with zero divisors in many different ways, by mixing and matching different notions of "irreducible " elements, "equivalent "... 展开 In recent decades, mathematicians have extended the definition of a unique factorization domain to rings with zero divisors in many different ways, by mixing and matching different notions of "irreducible " elements, "equivalent " factorizations, and "redundant factorizations " from the literature and deciding which elements are required to have "unique factorizations. " Their study raises many natural questions such as how many distinct kinds of "unique factorization rings " one can obtain through this approach, which combinations of choices lead to equivalent notions, and if one can find nice structural characterizations of each of these notions. This paper will be answering these questions. 收起