Absolute ideals of almost completely decomposable abelian groups

A ring is said to be a ring on an abelian group $G$, if its additive group coincides with the group $G$. A subgroup of the group $G$ is called the absolute ideal of $G$, if it is an ideal of every ring on the group $G$. If every ideal of a ring is an absolute ideal of its additive group, then the ring is called the $AI$-ring. If there exists at least one $AI$-ring on a group $G$, then the group $G$ is called the $RAI$-group. We consider rings on almost completely decomposable abealian groups (acd-groups) in the present paper.
A torsion free abelian group is an acd-group, if it contains a completely decomposable subgroup of finite rank and of finite index. Every acd-group $G$ contains the regulator $A$, which is completely decomposable and fully invariant. The finite quotient group $G/A$ is called the regulator quotient of the group $G$, the order of the group $G/A$ is called the regulator index. If the regulator quotient of an acd-group is cyclic, then the group is called the crq-group. If the types of the direct rank-1 summands of the regulator $A$ are pairwise incomparable, then the groups $A$ and $G$ are called rigid. If all these types are idempotent, then the group $G$ is of the ring type.
The main result of the present paper is that every rigid crq-group of the ring type is an $RAI$-group. Moreover, the principal absolute ideals are completely described for such groups.
Let $G$ be a rigid crq-group of the ring type. A subgroup $A$ is the regulator of the group $G$, the quotient $G/A=\langle d+A\rangle$ is the regulator quotient and $n$ is the regulator index. A decomposition $A=\bigoplus\limits_{\tau\in T(G)}A_\tau$ of the regulator $A$ into a direct sum of rank-1groups $A_\tau$ determines the set $T(G)=T(A)$ of critical types of the groups $A$ and $G$. Then for every $\tau\in T(G)$, there exists an element $e_\tau\in A_\tau$ such that $ A=\bigoplus\limits_{\tau\in T(G)} R_\tau e_\tau $, where $ R_\tau\, (\tau\in T(G))$ is a subring of the field of rational numbers containing the unit.
Moreover, the definition of natural near-isomorphism invariants $m_\tau \, (\tau\in$ $\in T(G))$ of the group $G$ naturally implies that every element $g\in G$ can be written in the divisible hull of the group $G$ in the following way $g=\!\!\!\sum\limits_{\tau\in T(G)}\cfrac{r_\tau}{m_\tau} e_\tau$, where $r_\tau$ are elements of the ring $R_\tau$ which are uniquely determined by a fixed decomposition of the regulator $A$.
Every description of RAI-groups is based on a description of principal absolute ideals of the groups. The least absolute ideal $\langle g\rangle_{AI}$ containing an element $g$ is called the principal absolute ideal generating by $g$. The following theorem describes principal absolute ideals.
Theorem 1. Let $G$ be a rigid crq-group of the ring type with a fixed decomposition of the regulator, $g=\sum\limits_{\tau\in T(G)}\cfrac{r_\tau}{m_\tau}e_\tau\in G$. Then
$$ \langle g\rangle_{AI}=\langle g\rangle+\bigoplus\limits_{\tau\in T(G)}{r_\tau}A_\tau. $$

Note that the elements $r_\tau\, (\tau\in T(G))$ in the representation of the element $g~\in~G$ are determined uniquely up to an invertible factor of $R_\tau$. Therefore, the representation of the principal absolute ideal doesn't depend on the decomposition of the regulator.
Theorem 2. Every rigid crq-group $G$ of the ring type is an $RAI$-group. In this case, for every integer $\alpha$ соprime to $n$ there exists an $AI$-ring $(G,\times)$ such that the equality $\overline{d}\times \overline{d}=\alpha \overline{d}$ takes place in the quotient ring $(G/A,\times)$, where $\overline{d}=d+A,G/A=\langle d\rangle$.
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