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

 $u\otimes v-v\otimes u$

where $u,v\in M$. Then the quotient algebra defined by

 $S(M):=T(M)/I$

is called the symmetric algebra over the ring $R$.

Remark. Let $R$ be a field, and $M$ a finite dimensional vector space over $R$. Suppose $\{e_{1},e_{2},\ldots,e_{n}\}$ is a basis of $M$ over $R$. Then $T(M)$ is nothing more than a free algebra on the basis elements $e_{i}$. Alternatively, the basis elements $e_{i}$ can be viewed as non-commuting indeterminates in the non-commutative polynomial ring $R\langle e_{1},e_{2},\ldots,e_{n}\rangle$. This then implies that $S(M)$ is isomorphic to the “commutative” polynomial ring $R[e_{1},e_{2},\ldots,e_{n}]$, where $e_{i}e_{j}=e_{j}e_{i}$.

Title symmetric algebra SymmetricAlgebra 2013-03-22 15:46:23 2013-03-22 15:46:23 CWoo (3771) CWoo (3771) 4 CWoo (3771) Definition msc 15A78