Torsion (Algebra): Abstract Algebra, Group (Mathematics), Periodic Group, Identity Element, Free Abelian Pure Subgroup, Finitely Generated Module, Analytic

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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.