Editing Symmetric algebra (section)

==Grading==
The symmetric algebra is a [[graded algebra]]. That is, it is a [[direct sum]]
:<math>S(V)=\bigoplus_{n=0}^\infty S^n(V),</math>
where <math>S^n(V),</math> called the {{mvar|n}}th [[symmetric power]] of {{mvar|V}}, is the vector subspace or submodule generated by the products of {{mvar|n}} elements of {{mvar|V}}. (The second symmetric power <math>S^2(V)</math> is sometimes called the '''symmetric square''' of {{mvar|V}}).

This can be proved by various means. One follows from the tensor-algebra construction: since the tensor algebra is graded, and the symmetric algebra is its quotient by a [[homogeneous ideal]]: the ideal generated by all <math>x \otimes y - y \otimes x,</math> where {{mvar|x}} and {{mvar|y}} are in {{mvar|V}}, that is, homogeneous of degree one.

In the case of a vector space or a free module, the gradation is the gradation of the polynomials by the [[total degree]]. A non-free module can be written as {{math|''L'' / ''M''}}, where {{mvar|L}} is a free module of base {{mvar|B}}; its symmetric algebra is the quotient of the (graded) symmetric algebra of {{mvar|L}} (a polynomial ring) by the homogeneous ideal generated by the elements of {{mvar|M}}, which are homogeneous of degree one.

One can also define <math>S^n(V)</math> as the solution of the universal problem for [[multilinear function|{{mvar|n}}-linear symmetric functions]] from {{mvar|V}} into a vector space or a module, and then verify that the [[direct sum]] of all <math>S^n(V)</math> satisfies the universal problem for the symmetric algebra.