|
In abstract algebra, the term torsion refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be free. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...
This picture illustrates how the hours on a clock form a group under modular addition. ...
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...
In mathematics, a free module is a module having a free basis. ...
Definition Let G be a group. An element g of G is called a torsion element if g has finite order. If all elements of G are torsion, then G is called a torsion group. If the only torsion element is the neutral element, then the group G is called torsion-free. This picture illustrates how the hours on a clock form a group under modular addition. ...
In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...
In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ...
In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...
Let M be a module over a ring R without zero divisors. An element m of M is called a torsion element if the cyclic submodule of M generated by m is not free. Equivalently, m is torsion if and only if it has a non-zero annihilator in R. 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). The module M is called a torsion module if T(M) = M, and is called torsion-free if T(M) = 0. If the ring R is non-commutative then the situation is more complicated, and the set of torsion elements need not be a submodule. Nevertheless, it is submodule under a very general assumption that the ring R satisfies the Ore condition. This covers the case when R is a Noetherian ring. In abstract algebra, a module is a generalization of a vector space. ...
In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. ...
In mathematics, a free module is a module having a free basis. ...
Annihilators are a concept that occurs in ring theory, a branch of mathematics. ...
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...
In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ...
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. In mathematics, an abelian group, also called a commutative group, is a group (G, * ) such that a * b = b * a for all a and b in G. In other words, the order in which the binary operation is performed doesnt matter. ...
The integers are commonly denoted by the above symbol. ...
More generally, let R be an arbitrary ring and S ⊂ R be a multiplicatively closed subset. Then one defines the notion of S-torsion as follows. An element m of an R-module M is called an S-torsion element if there exists s in S such that s annihilates m, i.e., sm = 0. In particular, one can take for S to be the set of all non-zero divisors of the ring R. In this case, S-torsion is frequently called simply torsion, extending the definition above from the case of domains to general rings. Annihilators are a concept that occurs in ring theory, a branch of mathematics. ...
In abstract algebra, a domain is the noncommutative analogue of an integral domain. ...
Examples 1. Let M be a free module over any ring R. Then it follows immediately from the definitions that M is torsion-free (if the ring R is not a domain then torsion is considered with respect to the set S of non-zero divisors of R). In particular, any free abelian group is torsion-free and any vector space over a field K is torsion-free when viewed as the module over K. In mathematics, a free module is a module having a free basis. ...
In abstract algebra, a free abelian group is an abelian group that has a basis in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
2. By contrast with Example 1, any finite group (abelian or not) is periodic and finitely generated. Burnside problem asks whether, conversely, any finitely generated periodic group must be finite. (The answer is "no" in general, even if the period is fixed.) In mathematics, a finite group is a group which has finitely many elements. ...
One of the oldest open problems in group theory was first posed by William Burnside in a paper published in 1902. ...
3. In the modular group, Γ obtained from the group SL(2,Z) of two by two integer matrices with unit determinant by factoring out its center, any nontrivial torsion element either has order two and is conjugate to the element S or has order three and is conjugate to the element ST. In this case, torsion elements do not form a subgroup, for example, S · ST = T, which has infinite order. In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...
4. The abelian group Q/Z, consisting of the rational numbers (mod 1), is periodic, i.e. every element has finite order. Analogously, the module K(t)/K[t] over the ring R = K[t] of polynomials in one variable is pure torsion. Both these examples can be generalized as follows: if R is a commutative domain and Q is its field of fractions, then Q/R is a torsion R-module. In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...
Case of a principal ideal domain Suppose that R is a (commutative) principal ideal domain and M is a finitely-generated R-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module M up to isomorphism. In particular, it claims that In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ...
In mathematics, a module is a finitely-generated module if it has a finite generating set. ...
// In abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain has two statements, which are equivalent by the Chinese remainder theorem: Every finitely generated module M over a principal ideal domain R is isomorphic to a unique one of the form where and . ...
 where F is a free R-module of finite rank (depeding only on M) and T(M) is the torsion submodule of M. As a corollary, any finite-generated torsion-free module over R is free. This corollary does not hold for more general commutative domains, even for R = K[x,y], the ring of polynomial in two variables.
Torsion and localization Assume that R is a commutative domain and M is an R-module. Let Q be the quotient field of the ring R. Then one can consider the Q-module In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. ...
 obtained from M by extension of scalars. Since Q is a field, a module over Q is a vector space, possibly, infinite-dimensional. There is a canonical homomorphism of abelian groups from M to MQ, and the kernel of this homomorphism is precisely the torsion submodule T(M). More generally, if S is a multiplicatively closed subset of the ring R, then we may consider localization of the R-module M, In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In mathematics, the localization of a module is a construction to introduce denominators in a module M for a ring R. It has become fundamental in particular in algebraic geometry, as the link between modules and sheaf theory. ...
 which is a module over the local ring RS. There is a canonical map from M to MS, whose kernel is precisely the S-torsion submodule of M. Thus the torsion submodule of M can be interpreted as the set of the elements that 'vanish in the localization'. The same interpretation continues to hold in the non-commutative setting for rings satisfying the Ore condition. In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...
Torsion in homological algebra The concept of torsion plays an important role in homological algebra. If M and N are two modules over a commutative ring R (for example, two abelian groups, when R = Z), Tor functors yield a family of R-modules Tori(M,N). Loosely speaking, nontrivial torsion in M can be detected by the the higher Tor functors (i greater than zero) with an appropriate module N, at least when the ring R is a domain. The symbol Tor denoting the functors reflects this relation with the algebraic torsion. Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...
In mathematics, the Tor functors of homological algebra are the derived functors of the tensor product functor. ...
In abstract algebra, a domain is the noncommutative analogue of an integral domain. ...
See also Some torsions is the name for sometopological invariants. ...
In mathematics, the localization of a module is a construction to introduce denominators in a module M for a ring R. It has become fundamental in particular in algebraic geometry, as the link between modules and sheaf theory. ...
In abstract algebra, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. ...
In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A. It shows that the integral homology groups Hi(X,Z) do in a certain, definite sense...
References - Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, ISBN 0-8176-3065-1
- Michiel Hazewinkel (2001), "Torsion submodule", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 1-55608-010-7
|
Feeds |
Contact