Algebra (ring theory)

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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.

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

Contents 1 Formal definition 2 Algebra homomorphisms 3 Examples 4 Constructions 5 See also

[edit] 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

• 

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

• 
• 

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 whose image lies in the center of A. The algebra A can then be thought of as an R-module by defining

for all r ∈ R and x ∈ A.

[edit] Algebra homomorphisms

An algebra homomorphism between two R-algebras is just an R-linear ring homomorphism. Explicitly, is an algebra homomorphism if

• 
• 
• 
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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.

[edit] 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 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[x₁, ..., x_n] is a commutative R-algebra. In fact, this is the free commutative R-algebra on the set {x₁, ..., 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⊗_ZA can be given the structure of an R-algebra by defining r·(s⊗a) = (rs⊗a). The functor which sends A to R⊗_ZA is left adjoint to the functor which sends an R-algebra to its underlying ring (forgetting the module structure).

[edit] 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 = (r1_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.

[edit] See also

• algebra over a field
• associative algebra
• commutative algebra