- Algebra (ring theory)
In

mathematics , specifically inring 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 acommutative 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 additiveabelian 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**(themonoidal 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"-linearring 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 thecoslice category "R"/**CRing**where**CRing**is thecategory 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 thatabelian 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", theendomorphism 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.

*Everypolynomial 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".

*Thetensor algebra of an "R"-module is naturally an "R"-algebra. The same is true for quotients such as the exterior andsymmetric algebra s. Categorically speaking, thefunctor which maps an "R"-module to its tensor algebra isleft 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" isleft 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 asubmodule 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 thequotient 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 afree product of "R"-algebras in a manner similar to the free product of groups. The free product is thecoproduct in the category of "R"-algebras.;Tensor products: The tensor product of two "R"-algebras is also an "R"-algebra in a natural way. Seetensor 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

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