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In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis. Being an abelian group means that it is a set together with an associative, commutative, and invertible binary operation. Conventionally, this operation is thought of as addition and its inverse is thought of as subtraction on the group elements. A basis is a subset of the elements such that every group element can be found by adding or subtracting a finite number of basis elements, and such that, for every group element, its expression as a linear combination of basis elements is unique. For instance, the integers under addition form a free abelian group with basis . Addition of integers is commutative, associative, and has subtraction as its inverse operation, each integer can be formed by using addition or subtraction to combine some number of copies of the number 1, and each integer has a unique representation as an integer multiple of the number 1. Integer lattices also form examples of free abelian groups. Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. The elements of a free abelian group with basis ''B'' may be represented by expressions of the form where each coefficient ''ai'' is a nonzero integer, each factor ''bi'' is a distinct basis element, and the sum has finitely many terms. These expressions, and the group elements they represent, are also known as formal sums over ''B''. Alternatively, the elements of a free abelian group be thought of as signed multisets containing finitely many elements of ''B'', with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent the elements of a free abelian group is as the functions from ''B'' to the integers that have finitely many nonzero values, with pointwise addition of these functions as the group operation. For every set ''B'' there is a free abelian group with ''B'' 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 element by element, a free group with basis ''B'' may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of ''B''. Alternatively, the free abelian group with basis ''B'' may be described by a presentation with the elements of ''B'' as its generators and with the commutators of pairs of members as its relators. Every free abelian group has a rank defined as the cardinality of a basis, every two bases for the same group give the same rank, and every two free abelian groups with the same rank are isomorphic. Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups. ==Examples and constructions== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Free abelian group」の詳細全文を読む スポンサード リンク
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