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An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field , and an operation called scalar multiplication between elements of the field (called scalars ), and elements of the vector space (called vectors ).
In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a ...
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".
In algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. The notion of algebraic structure has been formalized in universal algebra
An algebraic construction is a method by which an algebraic entity is defined or derived from another. Instances include: This list is incomplete; ...
Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a group and the pair of which together make it into a field. A measure: intervals of the real line have a specific length , which can be extended to the Lebesgue measure on many of its subsets .
A nonassociative ring is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras. [citation ...
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
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