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In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free -modules, the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors. The elements of a free abelian group with basis may be described in several equivalent ways. These include formal sums over , which are expressions of the form where each is a nonzero integer, each is a distinct basis element, and the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of , with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a function from to the integers with finitely many nonzero values; for this functional representation, the group operation is the pointwise addition of functions. Every set has a free abelian group with as its basis. This group is unique in the sense that every two free abelian groups with the same basis are isomorphic. Instead of constructing it by describing its individual elements, a free abelian group with basis may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of .

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Torsion (algebra)

In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsion submodule. A module is torsion-free if its torsion submodule comprises only the zero element. This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements.

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Finitely generated abelian group

In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that the set is a generating set of or that generate . Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. The integers, , are a finitely generated abelian group. The integers modulo , , are a finite (hence finitely generated) abelian group.

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