In mathematics, an associative algebra is a module which also allows the multiplication of vectors in a distributive and associative manner. They are thus a special case of algebras over commutative rings.
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.
If A itself is c...
more
In mathematics, an associative algebra is a module which also allows the multiplication of vectors in a distributive and associative manner. They are thus a special case of algebras over commutative rings.
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.
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.
This definition is equivalent to the statement that an R-algebra is a monoid in R-Mod (the monoidal category of R-modules).
Starting with a ring A, we get an associative R-algebra by providing a ring homomorphism whose image lies in the...
less
