Editing Associative algebra (section)

== 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 (mathematics)|ring]] ''A'' 
that is also an [[module (mathematics)|''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
: <math>r\cdot(xy) = (r\cdot x)y = x(r\cdot y)</math>
for all ''r'' in ''R'' and ''x'', ''y'' in the algebra. (This definition implies that the algebra, being a ring, is [[unital algebra|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 (ring theory)|center]] of ''A''. If ''f'' is such a homomorphism, the scalar multiplication is {{nowrap|(''r'', ''x'') ↦ ''f''(''r'')''x''}} (here the multiplication is the ring multiplication); if the scalar multiplication is given, the ring homomorphism is given by {{nowrap|''r'' ↦ ''r'' ⋅ 1<sub>''A''</sub>}}. (See also ''{{slink|#From ring homomorphisms}}'' below).

Every ring is an associative '''Z'''-algebra, where '''Z''' denotes the ring of the [[integer]]s.

A '''{{vanchor|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 (category theory)|monoid object]] in [[category of modules|'''''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
: <math>m : A \otimes_R A \to A</math>.
The associativity then refers to the identity:
: <math>m \circ ({\operatorname{id}} \otimes m) = m \circ (m \otimes \operatorname{id}).</math>

=== From ring homomorphisms ===
An associative algebra amounts to a [[ring homomorphism]] whose image lies in the [[center of a ring|center]]. Indeed, starting with a ring ''A'' and a ring homomorphism {{nowrap|''η'' : ''R'' → ''A''}} whose image lies in the [[center (ring theory)|center]] of ''A'', we can make ''A'' an ''R''-algebra by defining
: <math>r\cdot x = \eta(r)x</math>
for all {{nowrap|''r'' ∈ ''R''}} and {{nowrap|''x'' ∈ ''A''}}. If ''A'' is an ''R''-algebra, taking {{nowrap|1=''x'' = 1}}, the same formula in turn defines a ring homomorphism {{nowrap|''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 {{nowrap|''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 {{nowrap|''R'' → ''A''}} for a fixed ''R'', i.e., commutative ''R''-algebras, and whose morphisms are ring homomorphisms {{nowrap|''A'' → ''A''′}} that are under ''R''; i.e., {{nowrap|''R'' → ''A'' → ''A''′}} is {{nowrap|''R'' → ''A''′}} (i.e., the [[coslice category]] of the category of commutative rings under ''R''.)  The [[prime spectrum]] functor Spec then determines an [[dual (category theory)|anti-equivalence]] of this category to the category of [[affine scheme]]s 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]]''.