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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.
For a commutative ring R and commutative R-algebras A and B, Tor R * (A,B) has the structure of a graded-commutative algebra over R. Moreover, elements of odd degree in the Tor algebra have square zero, and there are divided power operations on the elements of positive even degree. [11]
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
A torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently rm = 0 implies r = 0 or m = 0. Noetherian A Noetherian module is a module that satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes ...
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.
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.
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.
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 ...