---- > [!definition] Definition. ([[symmetric algebra]]) > Let $M$ be a [[module]] over [[commutative ring]] $R$. We define the **symmetric algebra** of $M$ as the [[graded algebra|graded]] $R$-[[algebra]] $\bigoplus_{\ell \geq 0} \mathbb{S}_{R}^{\ell}(M),$ where multiplication is defined in terms of [[symmetric power|pure symmetric tensors]] by $(m_{1}\dots m_{i}) \cdot (n_{1}\dots,n_{j})=m_{1}\dots m_{i}n_{1}\dots n_{j}$ and is defined in general via the linear extension of this. ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```