Associative Algebra - Constructions

Constructions

Subalgebras

A subalgebra of an R-algebra A is a subset of A which is both a subring and a submodule of A. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of A.

Quotient algebras

Let A be an R-algebra. Any ring-theoretic ideal I in A is automatically an R-module since r·x = (r1_A)x. This gives the quotient ring A/I the structure of an R-module and, in fact, an R-algebra. It follows that any ring homomorphic image of A is also an R-algebra.

Direct products

The direct product of a family of R-algebras is the ring-theoretic direct product. This becomes an R-algebra with the obvious scalar multiplication.

Free products

One can form a free product of R-algebras in a manner similar to the free product of groups. The free product is the coproduct in the category of R-algebras.

Tensor products

The tensor product of two R-algebras is also an R-algebra in a natural way. See tensor product of algebras for more details.