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symmetric algebra

In mathematics, the symmetric algebra $S (V)$ (also denoted $Sym(V))$ on a vector space $V$ over a field $K$ is a commutative algebra over $K$ that contains $V$ , and is, in some sense, minimal for this property. Here, "minimal" means that $S (V)$ satisfies the following universal property: for every linear map $f$ from $V$ to a commutative algebra $A$ , there is a unique algebra homomorphism $g : S (V) \to A$ such that $f = g \circ i$ , where $i$ is the inclusion map of $V$ in $S (V)$ .
If $B$ is a basis of $V$ , the symmetric algebra $S (V)$ can be identified, through a canonical isomorphism, to the polynomial ring $K [B]$ , where the elements of $B$ are considered as indeterminates. Therefore, the symmetric algebra over $V$ can be viewed as a "coordinate free" polynomial ring over $V$ .
The symmetric algebra $S (V)$ can be built as the quotient of the tensor algebra $T (V)$ by the two-sided ideal generated by the elements of the form $x \otimes y - y \otimes x$ .
All these definitions and properties extend naturally to the case where $V$ is a module (not necessarily a free one) over a commutative ring.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Symmetric_algebra)

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