Some remarks on Pr\"ufer rings with zero-divisors

Abstract

Let $A$ be the fiber product $R\times_TB$, where $B\to T$ is a surjective ring homomorphism with regular kernel and $R\subseteq T$ is a ring extension where $T$ is an overring of $R$. In this paper we provide a characterization of when $A$ has distinguished Pr\"ufer-like properties and new constructions of Pr\"ufer rings with zero-divisors. Furthermore we give examples of homomorphic images of Pr\"ufer rings that are Pr\"ufer without assuming that the kernel of the surjection is regular. Finally we provide some remarks on the ideal theory of pre-Pr\"ufer rings.