In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, a torsion-free abelian group is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
which has no non-trivial
torsion elements; that is, a
group in which the
group operation is
commutative and the
identity element is the only element with finite
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
.
While
finitely generated abelian group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, ...
s are completely classified, not much is known about infinitely generated abelian groups, even in the torsion-free countable case.
Definitions
An
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
is said to be torsion-free if no element other than the identity
is of finite
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
. Explicitly, for any
, the only element
for which
is
.
A natural example of a torsion-free group is
, as only the integer 0 can be added to itself finitely many times to reach 0. More generally, the
free abelian group is torsion-free for any
. An important step in the proof of the classification of finitely generated abelian groups is that every such torsion-free group is isomorphic to a
.
A non-finitely generated countable example is given by the additive group of the polynomial ring