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In this article, all rings and algebras are assumed to be Unital and Associative .


FORMAL DEFINITION


Let ''R'' be a Commutative Ring . An ''R''-algebra is a set ''A'' which has the structure of both a Ring and an ''R''-module in such a way that ring multiplication is an ''R''- Bilinear Map . Explicity, we must have
  • r\cdot(xy) = (r\cdot x)y = x(r\cdot y)

    • If ''A'' itself is commutative (as a ring) then it is called a commutative ''R''-algebra.


Starting with an ''R''-module ''A'', we get an ''R''-algebra by equipping ''A'' with an ''R''- Bilinear Map ''A'' × ''A'' → ''A'' such that
  • x(yz) = (xy)z\,

  • \exists 1\in A,\; 1x = x1 = x

    • for all ''x'', ''y'', and ''z'' in ''A''. This ''R''-bilinear map then gives ''A'' the structure of a ring.


Conversely, starting with a ring ''A'', we get an ''R''-algebra by providing a Ring Homomorphism ho\colon R o A whose image lies in the Center of ''A''. The algebra ''A'' can then be thought of as an ''R''-module by defining
:r\cdot x = ho(r)x
for all ''r'' ∈ ''R'' and ''x'' ∈ ''A''.


ALGEBRA HOMOMORPHISMS


An ''algebra Homomorphism '' between two ''R''-algebras is just an ''R''-linear Ring Homomorphism . Explicity, \phi : A_1 o A_2 is an algebra homomorphism if
  • \phi(r\cdot x) = r\cdot \phi(x)

  • \phi(x+y) = \phi(x)+\phi(y)\,

  • \phi(xy) = \phi(x)\phi(y)\,

  • \phi(1) = 1\,

    • The class of all ''R''-algebras together with algebra homomorphisms between them form a Category , sometimes denoted ''R''-Alg.


The subcategory of commutative ''R''-algebras can be characterized as the Coslice Category ''R''/CRing where CRing is the category of commutative rings.


EXAMPLES


  • Any ring ''A'' can be considered as a Z-algebra in a unique way. 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 ''n'' is a (Z/''n''Z)-algebra in the same way.

  • Any ring ''A'' is an algebra over its Center ''Z''(''A''), or over any subring of its center.

  • Any commutative ring ''R'' is an algebra over itself, or any subring of ''R''.

  • Given an ''R''-module ''M'', the Endomorphism Ring of ''M'', denoted End''R''(''M'') is an ''R''-algebra by defining (''r''·φ)(''x'') = ''r''·φ(''x'').

  • Any ring of 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 ''R''-module.

  • Every Polynomial Ring ''R'' ..., ''x''''n'' is a commutative ''R''-algebra. In fact, this is the free commutative ''R''-algebra on the set {''x''1, ..., ''x''''n''}.

  • The 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 a naturally an ''R''-algebra. The same is true for quotients such as the Exterior and Symmetric Algebra s. Categorically speaking, the Functor which maps an ''R''-module to its tensor algebra is Left Adjoint to the functor which sends an ''R''-algebra to its underlying ''R''-module (forgetting the ring structure).

  • Given a commutative ring ''R'' and any ring ''A'' the Tensor Product ''R''⊗Z''A'' can be given the structure of an ''R''-algebra by defining ''r''·(''s''⊗''a'') = (''rs''⊗''a''). The functor which sends ''A'' to ''R''⊗Z''A'' is Left Adjoint to the functor which sends an ''R''-algebra to its underlying ring (forgetting the module structure).



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 ''I'' in ''A'' is automatically an ''R''-module since ''r''·''x'' = (''r''1''A'')''x''. This gives the Quotient Ring ''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 direct product. This becomes an ''R''-algebra with the obvious scalar multiplication.
;Free products: One can form a 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.


SEE ALSO