On quasi-arithmetic rings

Abstract

In this short note we characterize the rings (commutative with unity) that satisfy the following property:

(*) for every ideals \( I \) and \( J \) is \( I \cdot J = (I + J) \cdot (I \cap J) .\)

In the case of a domain we recover a well known characterization of Prüfer domains. Moreover we give a more explicit description in the case of a noetherian ring.