In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.
In this article we will also use the term K-algebra to mean an associative algebra over the field K.
In this article associative algebras are assumed to have a multiplicative unit, denoted 1; they are sometimes called unital associative algebras for clarification.
In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital.
Many authors consider the more general concept of an associative algebra over a commutative ring R, instead of a field: An R-algebra is an R-module with an associative R-bilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if S is any ring with center C, then S is an associative C-algebra.
Definition[επεξεργασία | επεξεργασία κώδικα]
Let R be a fixed commutative ring (so R could be a field). An associative R-algebra (or more simply, an R-algebra) is an additive abelian group A which has the structure of both a ring and an R-module in such a way that the scalar multiplication satisfies
for all r ∈ R and x, y ∈ A. Furthermore, A is assumed to be unital, which is to say it contains an element 1 such that
for all x ∈ A. Note that such an element 1 must be unique.
In other words, A is an R-module together with (1) an R-bilinear map A × A → A, called the multiplication, and (2) the multiplicative identity, such that the multiplication is associative:
for all x, y, and z in A. (Technical note: the multiplicative identity is a datum, while associativity is a property. By the uniqueness of the multiplicative identity, "unitarity" is often treated like a property.) If one drops the requirement for the associativity, then one obtains a non-associative algebra.
If A itself is commutative (as a ring) then it is called a commutative R-algebra.
- Put in another way, there is the forgetful functor from the category of unital associative algebras to the category of possibly non-unital associative algebras.
Εσωτερική Αρθρογραφία[επεξεργασία | επεξεργασία κώδικα]
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