In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.
In some areas of mathematics, associative algebras are typically assumed to have a multiplicative unit, denoted 1. To make this extra assumption clear, these associative algebras are called unital algebras.
Let R be a fixed commutative ring. An associative 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 ring multiplication is R-bilinear:
for all r ∈ R and x, y ∈ A. We say A is unital if it contains an element 1 such that
for all x ∈ A.
If A itself is commutative (as a ring) then it is called a commutative R-algebra.
Starting with an R-module A, we get an associative R-algebra by equipping A with an R-bilinear mapping A × A → A such that
for all x, y, and z in A. This R-bilinear mapping then gives A the structure of a ring and an associative R-algebra. Every associative R-algebra arises this way.
Moreover, the algebra A built this way will be unital if and only if
This definition is equivalent to the statement that a unital associative R-algebra is a monoid in R-Mod (the monoidal category of R-modules).
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