Algebra

In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R.

In this article, all rings are assumed to be unital.

Formal Definition

Let R be a commutative ring. An algebra is an R-module A together with a binary operation [·, ·]

called A-multiplication, which satisfies the following axiom:

• Bilinearity:

for all scalars a, b in R and all elements x, y, z in A.

Associative Algebras

If A is a monoid under A-multiplication (it satisfies associativity, identity, and totality), then the R-algebra is called an associative algebra. Interestingly, an associative algebra forms a ring over R and provides a generalization of a ring. An equivalent definition of an associative R-algebra is a ring homomorphism such that the image of f is contained in the centre of A.

See also

• associative algebra
• commutative algebra
• Lie algebra
• semiring

References

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4