Geodesic
On Finite Essential Extensions of Torsion Free Abelian Groups
Canadian journal of mathematics, Tome 48 (1996) no. 5, pp. 918-929

Voir la notice de l'article provenant de la source Cambridge University Press

Let A be a torsion free abelian group. We say that a group K is a finite essential extension of A if K contains an essential subgroup of finite index which is isomorphic to A. Such K admits a representation as (A Z x kx)/Zky where y = Nx + a for some k x k matrix N over Z and α ∈ A k satisfying certain conditions of relative primeness and Zk = {(α1,..., αk) : αi, ∈ Z}. The concept of absolute width of an f.e.e. K of A is defined and it is shown to be strictly smaller than the rank of A. A kind of basis substitution with respect to Smith diagonal matrices is shown to hold for homogeneous completely decomposable groups. This result together with general properties of our representations are used to provide a self contained proof that acd groups with two critical types are direct sum of groups of rank one and two.
DOI : 10.4153/CJM-1996-047-8
Mots-clés : 20K15, 20K35, decomposition, finite extension, representation, substitution, torsion free, width 
Benabdallah, K.; Ouldbeddi, M. A. On Finite Essential Extensions of Torsion Free Abelian Groups. Canadian journal of mathematics, Tome 48 (1996) no. 5, pp. 918-929. doi: 10.4153/CJM-1996-047-8
@article{10_4153_CJM_1996_047_8,
     author = {Benabdallah, K. and Ouldbeddi, M. A.},
     title = {On {Finite} {Essential} {Extensions} of {Torsion} {Free} {Abelian} {Groups}},
     journal = {Canadian journal of mathematics},
     pages = {918--929},
     year = {1996},
     volume = {48},
     number = {5},
     doi = {10.4153/CJM-1996-047-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1996-047-8/}
}
TY  - JOUR
AU  - Benabdallah, K.
AU  - Ouldbeddi, M. A.
TI  - On Finite Essential Extensions of Torsion Free Abelian Groups
JO  - Canadian journal of mathematics
PY  - 1996
SP  - 918
EP  - 929
VL  - 48
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1996-047-8/
DO  - 10.4153/CJM-1996-047-8
ID  - 10_4153_CJM_1996_047_8
ER  - 
%0 Journal Article
%A Benabdallah, K.
%A Ouldbeddi, M. A.
%T On Finite Essential Extensions of Torsion Free Abelian Groups
%J Canadian journal of mathematics
%D 1996
%P 918-929
%V 48
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1996-047-8/
%R 10.4153/CJM-1996-047-8
%F 10_4153_CJM_1996_047_8

[1] 1. Arnold, D.M., A class of pure subgroups of completely decomposable groups, Proc. Amer. Math., Soc. 41(1973), 37–44. Google Scholar

[2] 2. Arnold, D.M., Finite Rank Torsion free Abelian groups and Rings, Lecture Notes in Math. 931, Springer- Verlag, 1982. Google Scholar

[3] 3. Baer, R., Abelian groups without elements of finite order, Duke Math., J. 3(1937), 68–122. Google Scholar

[4] 4. Burkhardt, R., On a special class of almost completely decomposable groups I, Abelian Groups and Modules, Proceedings of the Udine Conference 1984, CISM courses and Lecture Notes 287, 141–150. Google Scholar

[5] 5. Fuchs, L., Infinite Abelian Groups, Vol. I, II, Academic Press, 1973. Google Scholar

[6] 6. Lady, L., Almost completely decomposable torsion free abelian groups, Proc. Amer. Math., Soc. 45(1974), 41–17. Google Scholar

[7] 7. Lady, L., Nearly isomorphic torsion free abelian groups, J., Algebra 35(1975), 235–238. Google Scholar

[8] 8. Lewis, W.S., Almost completely decomposable groups with two critical types, Comm., Algebra 21(1993), 607–614. Google Scholar

[9] 9. Mader, A., On the automorphism group and the endomorphism ring of abelian groups, Ann. Univ. Sci., Budapest 8(1965), 3–12. Google Scholar

[10] 10. Mader, A., Almost Completely Decomposable Abelian Groups, Montreal Notes, 1992. Google Scholar

Cité par Sources :