De dicto and de re

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 R-algebra is an R-module A together with a binary operation [·, ·]

$[\cdot ,\cdot ]:A\times A\to A$

called A-multiplication, which satisfies the following axiom:

• Bilinearity:

$[ax+by,z]=a[x,z]+b[y,z],\quad [z,ax+by]=a[z,x]+b[z,y]$

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 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 $f:R\to A$ such that the image of f is contained in the center of A.

Alternative definition: Given a ring homomorphism $\lambda :A\to B$ we say that B is an A-algebra. (Matsumura, Commutative Ring Theory, p 269.)

A ring homomorphism $\rho :A\to B$ 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 $\rho$. Every ring is a $\mathbb {Z}$-algebra. Kunz, Intro, Conventions.

See also

• Abelian algebra
• Algebraic structure (a much more general term)
• Associative algebra
• Coalgebra
• Graded algebra
• Lie algebra
• Semiring
• Split-biquaternion (example)
• Example of a non-associative algebra (example)

References

• Template:Lang Algebra