# 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.

## 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:

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

If the ring B is a commutative ring, a simpler, alternative definition is: 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.