Artinian modules over groups of rank 2

D.Nikitchenko

The problem of description of Artinian modules over the ring $ZG$, where $G$ is a free abelian group of rank $\le2$, is ``wild" from the viewpoint of the representation theory. But it is possible to describe some types of such modules. The study of Artinian modules over groups of rank 2 is initiated in this paper. In particular, we describe the structure of the injective hull of Artinian modules over the group ring $FG$, where $F$ is a field and $G$ is a free abelian group of rank 2.

1. Hartley B., McDougall D. Injective modules and solvable groups, satisfying the minimal condition for normal subgroups // Bull. Austral. Math. Soc. 4 (1971), 113–135.

2. Krugljak A. On representations of the group $(p,p)$ over a field of characteristic $p$ // Dok. Akad. Nauk. SSSR 156 (1968), 1253--1256. (in Russian)

3. Musson I. M. Injective modules for group rings of polycyclic groups // Quart. J. Math. Oxford 31 (1980), 429–466.

4. Wilson J. S. Some properties of groups inherited by normal subgroups of finite index // Math. Zametki. 114 (1970), 19–21.

5. Kudrachenko L. A. Artinian modules over groups of finite rank and the weak minimal condition for normal subgroups // Ricerche Mat. 44 (1995), 303–335.

6. Sharp R. J. Steps in commutative algebra, Cambridge Univ. Press: Cambridge, 1990.

7. Robinson D.j.S. Finiteness conditions and generalized solvable groups, Part 1, Springer: Berlin, 1972.

8. Matlis E. Injective modules over noetherian rings // Pacific J. Math. 3 (1958), no. 8, 511–528.

9. Robinson D.j.S. Finiteness conditions and generalized solvable groups, Part 2, Springer: Berlin, 1972.

10. Sharpe D. W., Vamos P. Injective modules, Cambridge Univ. Press: Cambridge, 1972.

	Artinian modules over groups of rank 2
Author	D.Nikitchenko Department of,Algebra, Faculty of Mechanics and Mathematics, Dnipropetrovsk State,University, 320625 Dnipropetrovsk, Provulok Naykovyi 13, Ukraine
Abstract	The problem of description of Artinian modules over the ring $ZG$, where $G$ is a free abelian group of rank $\le2$, is ``wild" from the viewpoint of the representation theory. But it is possible to describe some types of such modules. The study of Artinian modules over groups of rank 2 is initiated in this paper. In particular, we describe the structure of the injective hull of Artinian modules over the group ring $FG$, where $F$ is a field and $G$ is a free abelian group of rank 2.
Keywords	Artinian modules, group rings, free abelian groups of rank ≤2, representation theory, injective hull of modules, structure of Artinian modules, algebraic module theory, wild representation type
DOI	doi:10.30970/ms.14.2.141-149
Reference	1. Hartley B., McDougall D. Injective modules and solvable groups, satisfying the minimal condition for normal subgroups // Bull. Austral. Math. Soc. 4 (1971), 113–135. 2. Krugljak A. On representations of the group $(p,p)$ over a field of characteristic $p$ // Dok. Akad. Nauk. SSSR 156 (1968), 1253--1256. (in Russian) 3. Musson I. M. Injective modules for group rings of polycyclic groups // Quart. J. Math. Oxford 31 (1980), 429–466. 4. Wilson J. S. Some properties of groups inherited by normal subgroups of finite index // Math. Zametki. 114 (1970), 19–21. 5. Kudrachenko L. A. Artinian modules over groups of finite rank and the weak minimal condition for normal subgroups // Ricerche Mat. 44 (1995), 303–335. 6. Sharp R. J. Steps in commutative algebra, Cambridge Univ. Press: Cambridge, 1990. 7. Robinson D.j.S. Finiteness conditions and generalized solvable groups, Part 1, Springer: Berlin, 1972. 8. Matlis E. Injective modules over noetherian rings // Pacific J. Math. 3 (1958), no. 8, 511–528. 9. Robinson D.j.S. Finiteness conditions and generalized solvable groups, Part 2, Springer: Berlin, 1972. 10. Sharpe D. W., Vamos P. Injective modules, Cambridge Univ. Press: Cambridge, 1972.
Pages	141-149
Volume	14
Issue	2
Year	2000
Journal	Matematychni Studii
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