Absolute ideals of almost completely decomposable abelian groups

E. I. Kompantseva ; A. A. Fomin

Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 200-211.

Résumé

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 $\overline{d}=d+A,G/A=\langle d\rangle$. Bibliography: 16 titles.

Détail

Keywords: the ring on an abelian group, almost completely decomposable group, absolute ideal, $RAI$-group.

Bibliographie

[1] Fried E., “On the subgroups of abelian groups that ideals in every ring”, Proc. Colloq. Abelian Groups (Budapest, 1964), 51–55 | MR | Zbl

[2] Fuchs L., Infinite abelian groups, v. 2, Academic Press, New York–London, 1973, 416 pp. | MR | Zbl

[3] Beaumont R. A., Pierce R. S., “Torsion free rings”, Ill. J. Math., 5 (1961), 61–98 | MR | Zbl

[4] Beamount R. A., Lawver D. A., “Strongly semisimple abelian groups”, Publ. J. Math., 53:2 (1974), 327–336 | MR

[5] Gardner B. J., “Rings on completely decomposable torson-free abelian groups”, Comment. Math. Univ. Carolinae, 15:3 (1974), 381–382 | MR

[6] Kompantseva E. I., “Torsion free rings”, J. of Mathematical Sciences, 171:2 (2010), 213–247 | DOI | MR | Zbl

[7] Kompantseva E. I., “Absolute Nil-Ideals of Abelian Groups”, J. of Mathematical Sciences, 197:5 (2014), 625–634 | DOI | MR | Zbl

[8] Pham T. T. T., “Absolute ideals of abelian groups”, Abstraction of Southern Regional Algebra Conference (Montgomery, Alabama, 2010), 10

[9] Chehlov A. R., “On abelian groups where all subgroups are ideals”, Vestnik Tomskogo Gos. Univ., 2009, no. 3, 64–67

[10] McLean K. R., “The additive adeals of a $p$-ring”, J. London Math. Soc., 2 (1975), 523–529 | DOI | MR

[11] McLean K. R., “$p$-ring whose all right ideals are the fully invariant subgroups”, Proc. London Math. Soc., 3 (1975), 445–458 | DOI | MR | Zbl

[12] Mader A., Almost completely decomposable abelian groups, Algebra, Logic and Applications, 13, Gordon and Breach, Amsterdam, 1999

[13] Blagoveshchenskaya E. A., Almost completely decomposable abelian groups and their endomorphism rings, Izdatelstvo Politehnicheskogo universiteta, SPb., 2009

[14] Fomin A. A., “Quotient divisible and almost completely decomposable groups”, Models, Modules and Abelian Groups, in Memory of A. L. S. Corner, de Gruyter, Berlin–New York, 2008, 147–168 | MR

[15] Kompantseva E. I., “Rings on almost completely decomposable Abelian groups”, J. of Mathematical Sciences, 163:6 (2009), 688–693 | DOI | MR | Zbl

[16] Kompantseva E. I., “Multiplications on torsion free abelian groups of finite rank”, Algebra, theory of numbers and discrete geometry: modern problems and applications, Proc. XIII Int. conf. devoted to 85th anniversary of the birth of professor S. S. Rischkov, Izdatelstvo Tul. gos. ped. univ. im. L. N. Tolstogo, Tula, 2015, 79–81

[17] Blagoveshchenskaya E. A., “Almost completely decomposable groups and rings”, Journal of Math. Sci., 152:2 (2008), 137–154 | DOI | MR | Zbl

[18] Blagoveshchenskaya E. A., “Classification and realization theorems for one class of finite rank torsion-free rings”, Russian Mathematical Surveys, 61:4 (2006), 770–771 | DOI | MR | Zbl