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Torsion (Algebra)
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High Quality Content by WIKIPEDIA articles! In abstract algebra, the term torsion refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be free. Let G be a group. An element g of G is called a torsion element if g has finite order. If all elements of G are torsion, then G is called a torsion group. If the only torsion element is the identity element, then the group G is called torsion-free. Let M be a module over a ring R without zero divisors. An element m of M is called a torsion element if the cyclic submodule of M generated by m is not free. Equivalently, m is torsion if and only if it has a non-zero annihilator in R. If the ring R is commutative, then the set of all torsion elements forms a submodule of M, called the torsion submodule of M, sometimes denoted T(M). The module M is called a torsion module if T(M) = M, and is called torsion-free if T(M) = 0. If the ring R is non-commutative then the situation is more complicated, and the set of torsion elements need not be a submodule. Nevertheless, it is a submodule given the assumption that the ring R satisfies the Ore condition. This covers the case when R is a Noetherian domain.
Weitere Informationen
- Allgemeine Informationen
- GTIN 09786130349356
- Editor Lambert M. Surhone, Miriam T. Timpledon, Susan F. Marseken
- Sprache Englisch
- Größe H220mm x B150mm x T5mm
- Jahr 2010
- EAN 9786130349356
- Format Fachbuch
- ISBN 978-613-0-34935-6
- Titel Torsion (Algebra)
- Untertitel Abstract Algebra, Group (Mathematics), Periodic Group, Identity Element, Free Abelian Group, Pure Subgroup, Finitely Generated Module, Analytic Torsion
- Gewicht 136g
- Herausgeber Betascript Publishers
- Anzahl Seiten 80
- Genre Mathematik
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