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A module M over a ring R is called a torsion module if all its elements are torsion elements, and torsion-free if zero is the only torsion element. [1] 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).
Here are some of the basic properties and computations of Tor groups. [4]Tor R 0 (A, B) ≅ A ⊗ R B for any right R-module A and left R-module B.; Tor R i (A, B) = 0 for all i > 0 if either A or B is flat (for example, free) as an R-module.
the element xy is a product of two torsion elements, but has infinite order. The torsion elements in a nilpotent group form a normal subgroup. [2] Every finite abelian group is a torsion group. Not every torsion group is finite however: consider the direct sum of a countable number of copies of the cyclic group C 2; this is a torsion group ...
In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is torsion free if its torsion submodule contains only the zero element. In integral domains the regular elements of the ring are its nonzero
1. A torsion element of a module over a ring is an element annihilated by some regular element of the ring. 2. The torsion submodule of a module is the submodule of torsion elements. 3. A torsion-free module is a module with no torsion elements other than zero. 4. A torsion module is one all of whose elements are torsion elements. 5.
Any torsionless module over a domain is a torsion-free module, but the converse is not true, as Q is a torsion-free Z-module that is not torsionless. If R is a commutative ring that is an integral domain and M is a finitely generated torsion-free module then M can be embedded into R n, and hence M is torsionless.
Torsion of a square section bar Example of torsion mechanics. In the field of solid mechanics, torsion is the twisting of an object due to an applied torque. [1] [2] Torsion could be defined as strain [3] [4] or angular deformation, [5] and is measured by the angle a chosen section is rotated from its equilibrium position. [6]
The left R-module M is finitely generated if there exist a 1, a 2, ..., a n in M such that for any x in M, there exist r 1, r 2, ..., r n in R with x = r 1 a 1 + r 2 a 2 + ... + r n a n. The set {a 1, a 2, ..., a n} is referred to as a generating set of M in this case. A finite generating set need not be a basis, since it need not be linearly ...