AI-rings on almost completely decomposable Abelian groups

Abstract

We consider the class $A_0$ of reduced abelian block-rigid CRQ-groups of ring type. An absolute ideal of an abelian group G is a subgroup of G which is an ideal in any ring on G. A ring R is called an AI-ring if any ideal of R is an absolute ideal of its additive group. An abelian group G is called an RAI-group if there exists at least one AI-ring on G. It is proved that any group in $A_0$ is an RAI-group. Thus, in the class $A_0$, we solve Problem 93 in the monograph Fuchs (1973) [9]. We classify AI-rings on groups in the class $A_0$.