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.

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Formal Definition

Let R be a commutative ring. An 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 %2B b y, z] = a [x, z] %2B b [y, z], \quad [z, a x %2B b y] = a[z, x] %2B 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.

See also

References