Associative Algebra - Examples

Examples

• The square n-by-n matrices with entries from the field K form a unital associative algebra over K.
• The complex numbers form a 2-dimensional unital associative algebra over the real numbers.
• The quaternions form a 4-dimensional unital associative algebra over the reals (but not an algebra over the complex numbers, since if complex numbers are treated as a subset of the quaternions, complex numbers and quaternions do not commute).
• The 2 × 2 real matrices form an associative algebra useful in plane mapping.
• The polynomials with real coefficients form a unital associative algebra over the reals.
• Given any Banach space X, the continuous linear operators A : X → X form a unital associative algebra (using composition of operators as multiplication); this is a Banach algebra.
• Given any topological space X, the continuous real- or complex-valued functions on X form a real or complex unital associative algebra; here the functions are added and multiplied pointwise.
• An example of a non-unital associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.
• The Clifford algebras, which are useful in geometry and physics.
• Incidence algebras of locally finite partially ordered sets are unital· associative algebras considered in combinatorics.
• 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 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 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).