Algebra (ring theory)

In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field "K" is replaced by a commutative ring "R".

Any ring can be thought of as an algebra over the commutative ring of integers. Algebras over a commutative ring can, therefore, be thought of as generalizations of rings.

In this article, all rings and algebras are assumed to be unital and associative.

Formal definition

Let "R" be a fixed commutative ring. An "R"-algebra is an additive abelian group "A" which has the structure of both a ring and an "R"-module in such a way that ring multiplication is "R"-bilinear:
*$rcdot(xy)\; =\; (rcdot\; x)y\; =\; x(rcdot\; y)$for all "r" ∈ "R" and "x", "y" ∈ "A".

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

From "R"-modules

Starting with an "R"-module "A", we get an "R"-algebra by equipping "A" with an "R"-bilinear mapping "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 mapping then gives "A" the structure of a ring and an "R"-algebra.

This definition is equivalent to the statement that an "R"-algebra is a monoid in "R"-Mod (the monoidal category of "R"-modules).

From rings

Starting with a ring "A", we get an "R"-algebra by providing a ring homomorphism $etacolon\; R\; o\; A$ whose image lies in the center of "A". The algebra "A" can then be thought of as an "R"-module by defining:$rcdot\; x\; =\; eta(r)x$for all "r" ∈ "R" and "x" ∈ "A".

If "A" is commutative then the center of "A" is equal to "A", so that a commutative "R"-algebra can be defined simply as a homomorphism $etacolon\; R\; o\; A$ of commutative rings.

Algebra homomorphisms

A homomorphism between two "R"-algebras is 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 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.

ee also

*associative algebra
*commutative algebra
*semiring

References

*cite book | first = Serge | last = Lang | authorlink = Serge Lang | title = Algebra | publisher = Springer | location = New York | year = 2002 | edition = (Rev. 3rd ed.) | series = Graduate Texts in Mathematics 211 | isbn = 0-387-95385-X