Algebra (ring theory)

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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[edit]

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[edit]

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.

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