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.
called A-multiplication, which satisfies the following axiom:
- for all scalars a, b in R and all elements x, y, z in A.
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.
If the ring B is a commutative ring, a simpler, alternative definition is: Given a ring homomorphism we say that B is an A-algebra. (Matsumura, Commutative Ring Theory, p 269.)
A ring homomorphism shall always map the identity of A to the identity of B. We also say that B/A is an algebra over A given by . Every ring is a -algebra. Kunz, Intro, Conventions.
- Abelian algebra
- Algebraic structure (a much more general term)
- Associative algebra
- Graded algebra
- Lie algebra
- Split-biquaternion (example)
- Example of a non-associative algebra (example)