Editing Associative algebra (section)

== Constructions ==
; Subalgebras : A subalgebra of an ''R''-algebra ''A'' is a subset of ''A'' which is both a [[subring]] and a [[submodule]] of ''A''. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of ''A''.
; Quotient algebras : Let ''A'' be an ''R''-algebra. Any ring-theoretic [[ideal (ring theory)|ideal]] ''I'' in ''A'' is automatically an ''R''-module since {{nowrap|1=''r'' · ''x'' = (''r''1<sub>''A''</sub>)''x''}}. This gives the [[quotient ring]] {{nowrap|''A'' / ''I''}} the structure of an ''R''-module and, in fact, an ''R''-algebra. It follows that any ring homomorphic image of ''A'' is also an ''R''-algebra.
; Direct products : The direct product of a family of ''R''-algebras is the ring-theoretic [[product of rings|direct product]]. This becomes an ''R''-algebra with the obvious scalar multiplication.
; Free products: One can form a [[free product of associative algebras|free product]] of ''R''-algebras in a manner similar to the free product of groups. The free product is the [[coproduct]] in the category of ''R''-algebras.
; Tensor products : The tensor product of two ''R''-algebras is also an ''R''-algebra in a natural way. See [[tensor product of algebras]] for more details. Given a commutative ring ''R'' and any ring ''A'' the [[tensor product of rings|tensor product]] ''R''&nbsp;⊗<sub>'''Z'''</sub>&nbsp;''A'' can be given the structure of an ''R''-algebra by defining {{nowrap|1=''r'' · (''s'' ⊗ ''a'') = (''rs'' ⊗ ''a'')}}. The functor which sends ''A'' to {{nowrap|''R'' ⊗<sub>'''Z'''</sub> ''A''}} is [[left adjoint]] to the functor which sends an ''R''-algebra to its underlying ring (forgetting the module structure). See also: [[Change of rings]].
; Free algebra : A [[free algebra]] is an algebra generated by symbols. If one imposes commutativity; i.e., take the quotient by commutators, then one gets a polynomial algebra.