Editing Associative algebra (section)

=== 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]]''.