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Homesymmetric algebra

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# 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 $e_{i}e_{j}=e_{j}e_{i}$.

## Mathematics Subject Classification

15A78*no label found*

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