Weak, Strong and Very Strong Factorization

Abstract

An integral domain is said to have weak factorization if each nonzero nondivisorial ideal can be factored as the product of its divisorial closure and a finite product of (not necessarily distinct) maximal ideals. An integral domain is said to have strong factorization if it has weak factorization and the maximal ideals of the factorization are distinct. If, in addition, the maximal ideals in the factorization of a nonzero nondivisorial ideal I of the domain R can be restricted to those maximal ideals M such that IR M is not divisorial, we say that R has very strong factorization. In the present section, we study these properties with particular regard to the case of Prufer domains or almost Dedekind domains. In the Prufer case we provide several characterizations of domains having weak, strong or very strong factorization. We discuss the connections with h-local domains and we prove that very strong and strong factorizations are equivalent for Prufer domains.