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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).
A classical torsion wire-based du Noüy ring tensiometer. The arrow on the left points to the ring itself. The most common correction factors include Zuidema–Waters correction factors (for liquids with low interfacial tension), Huh–Mason correction factors (which cover a wider range than Zuidema–Waters), and Harkins–Jordan correction factors (more precise than Huh–Mason, while still ...
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 ...
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.
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 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.
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.
A nilpotent element in a nonzero ring is necessarily a zero divisor. An idempotent is an element such that e 2 = e. One example of an idempotent element is a projection in linear algebra. A unit is an element a having a multiplicative inverse; in this case the inverse is unique, and is denoted by a –1.