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Encyclopedia > Algebra over a ring

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. Euclid, detail from The School of Athens by Raphael. ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... This article presents the essential definitions. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

In this article, all rings and algebras are assumed to be unital and associative. In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ...

## Contents

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 In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...

• $rcdot(xy) = (rcdot x)y = x(rcdot y)$

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 × AA such that In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...

• $x(yz) = (xy)z,$
• $exists 1in A,; 1x = x1 = x$

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

Conversely, starting with a ring A, we get an R-algebra by providing a ring homomorphism $rhocolon R to A$ whose image lies in the center of A. The algebra A can then be thought of as an R-module by defining In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ...

$rcdot x = rho(r)x$

for all rR and xA.

## Algebra homomorphisms

An algebra homomorphism between two R-algebras is just an R-linear ring homomorphism. Explicity, $phi : A_1 to A_2$ is an algebra homomorphism if In abstract algebra, a homomorphism is a structure-preserving map. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...

• $phi(rcdot x) = rcdot phi(x)$
• $phi(x+y) = phi(x)+phi(y),$
• $phi(xy) = phi(x)phi(y),$
• $phi(1) = 1,$

The class of all R-algebras together with algebra homomorphisms between them form a category, sometimes denoted R-Alg. In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...

The subcategory of commutative R-algebras can be characterized as the coslice category R/CRing where CRing is the category of commutative rings. A comma category is a construction in category theory, a branch of mathematics. ...

## 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/nZ)-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 EndR(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[x1, ..., xn] is a commutative R-algebra. In fact, this is the free commutative R-algebra on the set {x1, ..., xn}.
• 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 RZA can be given the structure of an R-algebra by defining r·(sa) = (rsa). The functor which sends A to RZA is left adjoint to the functor which sends an R-algebra to its underlying ring (forgetting the module structure).

In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ... In abstract algebra, one associates to certain objects a ring, the objects endomorphism ring, which encodes several internal properties of the object. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, a free module is a module having a free basis. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In abstract algebra, a free algebra is the noncommutative analogue of a polynomial ring. ... In mathematics, the tensor algebra of a vector space V, denoted T(V) or Tâ€¢(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ... In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... In mathematics, the tensor product of two R-algebras is also an R-algebra in a natural way. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

## 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 = (r1A)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.

In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ... In abstract algebra, a module is a generalization of a vector space. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In abstract algebra, the free product of groups constructs a group from two or more given ones. ... In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... In mathematics, the tensor product of two R-algebras is also an R-algebra in a natural way. ...

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