ALGEBRA (RING THEORY)

In mathematics, specifically in ring theory, an 'algebra over a commutative ring' is a generalization of the concept of an associative algebra over a field, where the base field ''K'' is replaced by a commutative ring ''R''.
In this article, all rings and algebras are assumed to be unital and associative.

Contents

Formal definition

Algebra homomorphisms

Examples

Constructions

See also

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. Explicitly, we must have

★ $rcdot(xy)\; =\; (rcdot\; x)y\; =\; x(rcdot\; y)$
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

★ $x(yz)\; =\; (xy)z,$

★ $exists\; 1in\; A,;\; 1x\; =\; x1\; =\; x$
for all ''x'', ''y'', and ''z'' in ''A''. This ''R''-bilinear map then gives ''A'' the structure of a ring.
Defining an ''R''-algebra structure on ''A'' whose ''R''-module structure is the same as the one originally given is equivalent to giving an operation × satisfying the above conditions.
Starting with a ring ''A'', we get an ''R''-algebra by providing a ring homomorphism $$
hocolon R o A whose image lies in the center of ''A''. The algebra ''A'' can then be thought of as an ''R''-module by defining
:$rcdot\; x\; =$
ho(r)x
for all ''r'' ∈ ''R'' and ''x'' ∈ ''A''. Defining an ''R''-algebra structure on ''A'' preserving the original ring structure on A is then equivalent to giving a homomorphism $$
ho as above.

Algebra homomorphisms

An ''algebra homomorphism'' between two ''R''-algebras is just an ''R''-linear ring homomorphism. Explicitly, $phi\; :\; A\_1\; o\; A\_2$ is an algebra homomorphism if

★ $phi(rcdot\; x)\; =\; rcdot\; phi(x)$

★ $phi(x+y)\; =\; phi(x)+phi(y),$

★ $phi(xy)\; =\; phi(x)phi(y),$

★ $phi(1)\; =\; 1,$
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 groups 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''₁, ..., ''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 algebras. 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

★ algebra over a field

★ associative algebra

★ commutative algebra

★ semiring