symmetric algebra

Let M be a module over a commutative ring R. Form the tensor algebra T(M) over R. Let I be the ideal of T(M) generated by elements of the form


where u,vM. Then the quotient algebraPlanetmathPlanetmath defined by


is called the symmetric algebra over the ring R.

Remark. Let R be a field, and M a finite dimensional vector spaceMathworldPlanetmath over R. Suppose {e1,e2,,en} is a basis of M over R. Then T(M) is nothing more than a free algebraMathworldPlanetmath on the basis elements ei. Alternatively, the basis elements ei can be viewed as non-commuting indeterminates in the non-commutative polynomial ringMathworldPlanetmath Re1,e2,,en. This then implies that S(M) is isomorphicPlanetmathPlanetmathPlanetmath to the “commutativePlanetmathPlanetmath” polynomial ring R[e1,e2,,en], where eiej=ejei.

Title symmetric algebra
Canonical name SymmetricAlgebra
Date of creation 2013-03-22 15:46:23
Last modified on 2013-03-22 15:46:23
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 4
Author CWoo (3771)
Entry type Definition
Classification msc 15A78