Algebra (ring theory)

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.

[edit] 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.

[edit] Associative Algebras

If A is a monoid under A-multiplication (it satisfies associativity and it has an identity), then the R-algebra is called an associative algebra. 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 center of A.

[edit] See also

• associative algebra
• commutative algebra
• Lie algebra
• semiring
• split-biquaternion (example)
• Example of a non-associative algebra (example)

[edit] References

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