
This article is cited in 1 scientific paper (total in 1 paper)
Absolute ideals of almost completely decomposable abelian groups
E. I. Kompantseva^{ab}, A. A. Fomin^{b} ^{a} Financial University under the Government of the Russian Federation, Moscow
^{b} Moscow State Pedagogical University
Abstract:
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 (acdgroups) in the present paper.
A torsion free abelian group is an acdgroup, if it contains a
completely decomposable subgroup of finite rank and of finite
index. Every acdgroup $G$ 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 acdgroup is cyclic, then the group is
called the crqgroup. If the types of the direct rank1 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 crqgroup of the ring type is an $RAI$group. Moreover, the principal absolute ideals are completely described for such groups.
Let $G$ be a rigid crqgroup 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 rank1groups $A_\tau$ 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 nearisomorphism 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 RAIgroups 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 crqgroup 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 crqgroup $G$ of the ring type is an $RAI$group. In this case, for every integer $\alpha$ ñîprime 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 $\overline{d}=d+A,G/A=\langle d\rangle$.
Bibliography: 16 titles.
Keywords:
the ring on an abelian group, almost completely decomposable group, absolute ideal, $RAI$group.
Full text:
PDF file (254 kB)
References:
PDF file
HTML file
UDC:
512.541 Received: 09.11.2015
Citation:
E. I. Kompantseva, A. A. Fomin, “Absolute ideals of almost completely decomposable abelian groups”, Chebyshevskii Sb., 16:4 (2015), 200–211
Citation in format AMSBIB
\Bibitem{KomFom15}
\by E.~I.~Kompantseva, A.~A.~Fomin
\paper Absolute ideals of almost completely decomposable abelian groups
\jour Chebyshevskii Sb.
\yr 2015
\vol 16
\issue 4
\pages 200211
\mathnet{http://mi.mathnet.ru/cheb442}
\elib{https://elibrary.ru/item.asp?id=25006100}
Linking options:
http://mi.mathnet.ru/eng/cheb442 http://mi.mathnet.ru/eng/cheb/v16/i4/p200
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
This publication is cited in the following articles:

E. I. Kompantseva, A. A. Fomin, “Faktorno delimye gruppy i gruppy bez krucheniya, sootvetstvuyuschie konechnym abelevym gruppam”, Chebyshevskii sb., 20:2 (2019), 221–233

Number of views: 
This page:  190  Full text:  60  References:  59 
