On $\ddot{I}•$-Prüfer like conditions

Abstract

In this paper, we investigate the question of when a $\ddot{I}•$-ring is $\ddot{I}•$-Prüfer using two types of techniques: first, by analysing the lattice structure of the nonnil ideals of $\ddot{I}•$-rings; and secondly, by considering content ideal techniques which were developed to study Gaussian polynomials. In particular, we conclude that every Gaussian $\ddot{I}•$-ring is $\ddot{I}•$-Prüfer. Key concepts such as $\ddot{I}•$-weak global dimension, primary ideals and irreducible ideals are discussed, along with their hereditary properties in $\ddot{I}•$-Prüfer rings. We also prove that any semi-local $\ddot{I}•$-Prüfer ring is a $\ddot{I}•$-Bézout ring. This paper includes several theorems and examples that provide insights into the $\ddot{I}•$-Prüfer rings and their implications in the field of ring theory.