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 $\{e_1,e_2,\mathrm{\dots },e_n\}$ 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 $R⟨e_1,e_2,\mathrm{\dots },e_n⟩$. This then implies that $S(M)$ is isomorphic to the “commutative” polynomial ring $R[e_1,e_2,\mathrm{\dots },e_n]$, where $e_i{e}_j=e_j{e}_i$.