Serial factorizations of right ideals

Abstract

In a Dedekind domain $D$, every non-zero proper ideal $A$ factors as a product $A=P_1^{t_1}\cdots P_k^{t_k}$ of powers of distinct prime ideals $P_i$. For a Dedekind domain $D$, the $D$-modules $D/P_i^{t_i}$ are uniserial. We extend this property studying suitable factorizations $A=A_1\dots A_n$ of a right ideal $A$ of an arbitrary ring $R$ as a product of proper right ideals $A_1,\dots,A_n$ with all the modules $R/A_i$ uniserial modules. When such factorizations exist, they are unique up to the order of the factors. Serial factorizations turn out to have connections with the theory of $h$-local Pr\"ufer domains and that of semirigid commutative GCD domains.