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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 ...
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.
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
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 ...
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 ...
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.