Search results
Results from the WOW.Com Content Network
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.
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
In mathematics, the annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of S. Over an integral domain , a module that has a nonzero annihilator is a torsion module , and a finitely generated torsion module has a nonzero annihilator.
M is f.g. if and only if J(M) is a superfluous submodule of M, and M/J(M) is f.g. M is f.cog. if and only if soc(M) is an essential submodule of M, and soc(M) is f.g. If M is a semisimple module (such as soc(N) for any module N), it is f.g. if and only if f.cog. If M is f.g. and nonzero, then M has a maximal submodule and any quotient module M ...
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 ...
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.
Thus a module that contains nonzero torsion elements is not flat. In particular Q / Z {\displaystyle \mathbb {Q} /\mathbb {Z} } and all fields of positive characteristics are non-flat Z {\displaystyle \mathbb {Z} } -modules, where Z {\displaystyle \mathbb {Z} } is the ring of integers, and Q {\displaystyle \mathbb {Q} } is the field of the ...