CentralNotice From Wikipedia, the free encyclopedia Jump to: navigation , search 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 . 1 Formal definition 2 Example 2.1 Split-biquaternions 3 Associative algebras 4 Non-associative algebras 5 See also 6 References 7 Further reading Formal definition [ edit ] Let R be a commutative ring. An R -algebra is an R -module A together with a binary operation [·, ·] called A - multiplication , which satisfies the following axiom: Bilinearity : for all scalars , in R and all elements x , y , z in A . Example [ edit ] Split-biquaternions [ edit ] The split-biquaternions are an example of a...