On decomposable pseudofree groups
An Abelian group is pseudofree of rank ℓ if it belongs to the extended genus of ℤℓ, i.e., its localization at every prime p is isomorphic to ℤpℓ. A pseudofree group can be studied through a sequence of rational matrices, the so-called sequential representation. Here, we use these sequential represen...
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Wiley
1999-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171299226178 |
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| author | Dirk Scevenels |
| author_facet | Dirk Scevenels |
| author_sort | Dirk Scevenels |
| collection | DOAJ |
| description | An Abelian group is pseudofree of rank ℓ if it belongs to the extended genus of ℤℓ, i.e., its localization at every prime p is isomorphic to ℤpℓ. A pseudofree group can be studied through a sequence of rational matrices, the so-called sequential representation. Here, we use these sequential representations to study the relation between the product of extended genera of free Abelian groups and the extended genus of their direct sum. In particular, using sequential representations, we give a new proof of a result by Baer, stating
that two direct sum decompositions into rank one groups of a completely decomposable pseudofree Abelian group are necessarily equivalent. On the other hand, sequential representations can also be used to exhibit examples of pseudofree groups having nonequivalent direct sum decompositions into indecomposable groups. However, since this cannot occur when using the notion of near-isomorphism rather than isomorphism, we conclude our work by giving a characterization of near-isomorphism for pseudofree groups in terms of their sequential representations. |
| format | Article |
| id | doaj-art-730a90d91dca4b4fa644bdf4f5100216 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1999-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-730a90d91dca4b4fa644bdf4f51002162025-02-03T05:51:20ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251999-01-0122361762810.1155/S0161171299226178On decomposable pseudofree groupsDirk Scevenels0Centre de Recerca Matemàtica, Apartat 50, Bellaterra E-08193, SpainAn Abelian group is pseudofree of rank ℓ if it belongs to the extended genus of ℤℓ, i.e., its localization at every prime p is isomorphic to ℤpℓ. A pseudofree group can be studied through a sequence of rational matrices, the so-called sequential representation. Here, we use these sequential representations to study the relation between the product of extended genera of free Abelian groups and the extended genus of their direct sum. In particular, using sequential representations, we give a new proof of a result by Baer, stating that two direct sum decompositions into rank one groups of a completely decomposable pseudofree Abelian group are necessarily equivalent. On the other hand, sequential representations can also be used to exhibit examples of pseudofree groups having nonequivalent direct sum decompositions into indecomposable groups. However, since this cannot occur when using the notion of near-isomorphism rather than isomorphism, we conclude our work by giving a characterization of near-isomorphism for pseudofree groups in terms of their sequential representations.http://dx.doi.org/10.1155/S0161171299226178Localizationextended genustorsion-free Abelian group of finite ranksequential representationdirect sum decompositionnear-isomorphism. |
| spellingShingle | Dirk Scevenels On decomposable pseudofree groups International Journal of Mathematics and Mathematical Sciences Localization extended genus torsion-free Abelian group of finite rank sequential representation direct sum decomposition near-isomorphism. |
| title | On decomposable pseudofree groups |
| title_full | On decomposable pseudofree groups |
| title_fullStr | On decomposable pseudofree groups |
| title_full_unstemmed | On decomposable pseudofree groups |
| title_short | On decomposable pseudofree groups |
| title_sort | on decomposable pseudofree groups |
| topic | Localization extended genus torsion-free Abelian group of finite rank sequential representation direct sum decomposition near-isomorphism. |
| url | http://dx.doi.org/10.1155/S0161171299226178 |
| work_keys_str_mv | AT dirkscevenels ondecomposablepseudofreegroups |