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A group is called a torsion (or periodic) group if all its elements are torsion elements, and a torsion-free group if its only torsion element is the identity element. Any abelian group may be viewed as a module over the ring Z of integers, and in this case the two notions of torsion coincide.
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
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 torsion subgroup of an abelian group A is the subgroup of A that consists of all elements that have finite order. A torsion abelian group is an abelian group in which every element has finite order. A torsion-free abelian group is an abelian group in which the identity element is the only element with finite order.
The second polar moment of area, also known (incorrectly, colloquially) as "polar moment of inertia" or even "moment of inertia", is a quantity used to describe resistance to torsional deformation (), in objects (or segments of an object) with an invariant cross-section and no significant warping or out-of-plane deformation. [1]
The torsion constant or torsion coefficient is a geometrical property of a bar's cross-section. It is involved in the relationship between angle of twist and applied torque along the axis of the bar, for a homogeneous linear elastic bar. The torsion constant, together with material properties and length, describes a bar's torsional stiffness.
[2] On a local ring every finitely generated flat module is free. [3] A finitely generated flat module that is not projective can be built as follows. Let = be the set of the infinite sequences whose terms belong to a fixed field F. It is a commutative ring with addition and multiplication defined componentwise.