Basic concepts[edit]

Algebra homomorphisms[edit]

Given $K$ -algebras $A$ and $B$ , a homomorphism of $K$ -algebras or $K$ -algebra homomorphism is a $K$ -linear map $f : A \to B$ such that $f (xy) = f (x) f (y)$ for all $x, y$ in $A$ . The space of all $K$ -algebra homomorphisms between $A$ and $B$ is frequently written as

the algebra of all n-by-n over a field (or commutative ring) K. Here the multiplication is ordinary matrix multiplication.

matrices

where a group serves as a basis of the vector space and algebra multiplication extends group multiplication.

group algebras

the commutative algebra K[x] of all over K (see polynomial ring).

polynomials

algebras of , such as the R-algebra of all real-valued continuous functions defined on the interval [0,1], or the C-algebra of all holomorphic functions defined on some fixed open set in the complex plane. These are also commutative.

functions

are built on certain partially ordered sets.

Incidence algebras

algebras of , for example on a Hilbert space. Here the algebra multiplication is given by the composition of operators. These algebras also carry a topology; many of them are defined on an underlying Banach space, which turns them into Banach algebras. If an involution is given as well, we obtain B*-algebras and C*-algebras. These are studied in functional analysis.

linear operators

Algebra over an operad

Alternative algebra

Clifford algebra

Differential algebra

Free algebra

Geometric algebra

Max-plus algebra

Mutation (algebra)

Operator algebra

Zariski's lemma

; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004). Algebras, rings and modules. Vol. 1. Springer. ISBN 1-4020-2690-0.

distributivity