Editing Associative algebra (section)

=== Algebra ===
* Any ring ''A'' can be considered as a '''Z'''-algebra. The unique ring homomorphism from '''Z''' to ''A'' is determined by the fact that it must send 1 to the identity in ''A''. Therefore, rings and '''Z'''-algebras are equivalent concepts, in the same way that [[abelian group]]s and '''Z'''-modules are equivalent.
* Any ring of [[characteristic (algebra)|characteristic]] ''n'' is a ('''Z'''/''n'''''Z''')-algebra in the same way.
* Given an ''R''-module ''M'', the [[endomorphism ring]] of ''M'', denoted End<sub>''R''</sub>(''M'') is an ''R''-algebra by defining {{nowrap|1=(''r''·''φ'')(''x'') = ''r''·''φ''(''x'')}}.
* Any ring of [[matrix (mathematics)|matrices]] with coefficients in a commutative ring ''R'' forms an ''R''-algebra under matrix addition and multiplication. This coincides with the previous example when ''M'' is a finitely-generated, [[free module|free]] ''R''-module.
** In particular, the square ''n''-by-''n'' [[square matrix|matrices]] with entries from the field ''K'' form an associative algebra over ''K''.
* The [[complex number]]s form a 2-dimensional commutative algebra over the [[real number]]s.
* The [[quaternion]]s form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions).
* Every [[polynomial ring]] {{nowrap|''R''[''x''<sub>1</sub>, ..., ''x<sub>n</sub>'']}} is a commutative ''R''-algebra. In fact, this is the free commutative ''R''-algebra on the set {{nowrap|{{mset|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}}}}.
* The [[free algebra|free ''R''-algebra]] on a set ''E'' is an algebra of "polynomials" with coefficients in ''R'' and noncommuting indeterminates taken from the set ''E''.
* The [[tensor algebra]] of an ''R''-module is naturally an associative ''R''-algebra. The same is true for quotients such as the [[exterior algebra|exterior]] and [[symmetric algebra]]s. Categorically speaking, the [[functor]] that maps an ''R''-module to its tensor algebra is [[left adjoint]] to the functor that sends an ''R''-algebra to its underlying ''R''-module (forgetting the multiplicative structure).
* Given a module ''M'' over a commutative ring ''R'', the direct sum of modules {{nowrap|1=''R'' ⊕ ''M''}} has a structure of an ''R''-algebra by thinking ''M'' consists of infinitesimal elements; i.e., the multiplication is given as {{nowrap|1=(''a'' + ''x'')(''b'' + ''y'') = ''ab'' + ''ay'' + ''bx''}}. The notion is sometimes called the [[algebra of dual numbers]].
* A [[quasi-free algebra]], introduced by Cuntz and Quillen, is a sort of generalization of a free algebra and a semisimple algebra over an algebraically closed field.