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.

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.

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[edit] 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:

  • r\cdot(xy) = (r\cdot x)y = x(r\cdot 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.

[edit] 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 1\in 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).

[edit] From rings

Starting with a ring A, we get an R-algebra by providing a ring homomorphism \eta\colon 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

r\cdot 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 \eta\colon R \to A of commutative rings.

[edit] Algebra homomorphisms

A homomorphism between two R-algebras is an R-linear ring homomorphism. Explicitly, \phi : A_1 \to A_2 is an algebra homomorphism if

  • \phi(r\cdot x) = r\cdot \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.

[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 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 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 = (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.

[edit] See also

[edit] References