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 rยทx = (r1A)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.
Read more about this topic: Associative Algebra