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  2. Commutative property - Wikipedia

    en.wikipedia.org/wiki/Commutative_property

    Today the commutative property is a well-known and basic property used in most branches of mathematics. The first recorded use of the term commutative was in a memoir by François Servois in 1814, [ 1 ] [ 10 ] which used the word commutatives when describing functions that have what is now called the commutative property.

  3. Abelian group - Wikipedia

    en.wikipedia.org/wiki/Abelian_group

    A typical example is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian ...

  4. Tor functor - Wikipedia

    en.wikipedia.org/wiki/Tor_functor

    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. If R is an algebra over a ring S (which means in particular that S is commutative), then Tor R i (A, B) is at least an S-module.

  5. Field (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Field_(mathematics)

    For example, the integers Z form a commutative ring, but not a field: the reciprocal of an integer n is not itself an integer, unless n = ±1. In the hierarchy of algebraic structures fields can be characterized as the commutative rings R in which every nonzero element is a unit (which means every element is invertible).

  6. Proofs involving the addition of natural numbers - Wikipedia

    en.wikipedia.org/wiki/Proofs_involving_the...

    The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which has been proved above: a + 0 = a = 0 + a. Next we will prove the base case b = 1, that 1 commutes with everything, i.e. for all natural numbers a, we have a + 1 = 1 + a.

  7. Convolution - Wikipedia

    en.wikipedia.org/wiki/Convolution

    The symmetry of is the reason and are identical in this example. In mathematics (in particular, functional analysis ), convolution is a mathematical operation on two functions ( f {\displaystyle f} and g {\displaystyle g} ) that produces a third function ( f ∗ g {\displaystyle f*g} ).

  8. Commutator - Wikipedia

    en.wikipedia.org/wiki/Commutator

    In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory . Group theory

  9. Ring (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Ring_(mathematics)

    For example, if R is a commutative ring and f an element in R, then the localization [] consists of elements of the form /,, (to be precise, [] = [] / ().) [42] The localization is frequently applied to a commutative ring R with respect to the complement of a prime ideal (or a union of prime ideals) in R .