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To qualify as an abelian group, the set and operation, (,), must satisfy four requirements known as the abelian group axioms (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation is defined for any ordered pair of elements of A, that the result is well-defined, and that the ...
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry ...
If the operation additionally has an identity element, we have a commutative monoid; An abelian group, or commutative group is a group whose group operation is commutative. [16] A commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.) [18] In a field both addition and multiplication are ...
Every set can be the basis of a free abelian group, which is unique up to group isomorphisms. The free abelian group for a given basis set can be constructed in several different but equivalent ways: as a direct sum of copies of the integers, as a family of integer-valued functions, as a signed multiset, or by a presentation of a group.
In particular, the category of finitely generated modules over a noetherian commutative ring is abelian; in this way, abelian categories show up in commutative algebra. As special cases of the two previous examples: the category of vector spaces over a fixed field k is abelian, as is the category of finite-dimensional vector spaces over k.
A nonzero commutative ring in which every nonzero element has a multiplicative inverse is called a field. The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms. [4]
Moreover, for a fixed ring R, Tor is a functor in each variable (from R-modules to abelian groups). For a commutative ring R and R-modules A and B, Tor R i (A, B) is an R-module (using that A ⊗ R B is an R-module in this case). For a non-commutative ring R, Tor R i (A, B) is only an abelian group, in general.
In other words, / is abelian if and only if contains the commutator subgroup of . So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is.