In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K). The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a module or vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over K. A standard first example of a K-algebra is a ring of square matrices over a commutative ring K, with the usual matrix multiplication.

A commutative algebra is an associative algebra for which the multiplication is commutative, or, equivalently, an associative algebra that is also a commutative ring.

In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital.

Every ring is an associative algebra over its center and over the integers.

Definition

Let R be a commutative ring (so R could be a field). An associative R-algebra A (or more simply, an R-algebra A) is a ring A that is also an R-module in such a way that the two additions (the ring addition and the module addition) are the same operation, and scalar multiplication satisfies

${\displaystyle r\cdot (xy)=(r\cdot x)y=x(r\cdot y)}$

for all r in R and x, y in the algebra. (This definition implies that the algebra, being a ring, is unital, since rings are supposed to have a multiplicative identity.)

Equivalently, an associative algebra A is a ring together with a ring homomorphism from R to the center of A. If f is such a homomorphism, the scalar multiplication is (r, x) ↦ f(r)x (here the multiplication is the ring multiplication); if the scalar multiplication is given, the ring homomorphism is given by r ↦ r ⋅ 1_A. (See also § From ring homomorphisms below).

Every ring is an associative Z-algebra, where Z denotes the ring of the integers.

A commutative algebra is an associative algebra that is also a commutative ring.

As a monoid object in the category of modules

The definition is equivalent to saying that a unital associative R-algebra is a monoid object in R-Mod (the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the category of modules.

Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra A. For example, the associativity can be expressed as follows. By the universal property of a tensor product of modules, the multiplication (the R-bilinear map) corresponds to a unique R-linear map

${\displaystyle m:A\otimes _{R}A\to A}$.

The associativity then refers to the identity:

${\displaystyle m\circ ({\operatorname {id} }\otimes m)=m\circ (m\otimes \operatorname {id} ).}$

From ring homomorphisms

An associative algebra amounts to a ring homomorphism whose image lies in the center. Indeed, starting with a ring A and a ring homomorphism η : R → A whose image lies in the center of A, we can make A an R-algebra by defining

${\displaystyle r\cdot x=\eta (r)x}$

for all r ∈ R and x ∈ A. If A is an R-algebra, taking x = 1, the same formula in turn defines a ring homomorphism η : R → A whose image lies in the center.

If a ring is commutative then it equals its center, so that a commutative R-algebra can be defined simply as a commutative ring A together with a commutative ring homomorphism η : R → A.

The ring homomorphism η appearing in the above is often called a structure map. In the commutative case, one can consider the category whose objects are ring homomorphisms R → A for a fixed R, i.e., commutative R-algebras, and whose morphisms are ring homomorphisms A → A′ that are under R; i.e., R → A → A′ is R → A′ (i.e., the coslice category of the category of commutative rings under R.) The prime spectrum functor Spec then determines an anti-equivalence of this category to the category of affine schemes over Spec R.

How to weaken the commutativity assumption is a subject matter of noncommutative algebraic geometry and, more recently, of derived algebraic geometry. See also: Generic matrix ring.

Algebra homomorphisms

A homomorphism between two R-algebras is an R-linear ring homomorphism. Explicitly, φ : A₁ → A₂ is an associative algebra homomorphism if

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