---- > [!definition] Definition. ([[group algebra]]) > Let $G$ be a finite [[group]]. Let $R$ be a (say) [[commutative ring]]. > The **group algebra** or **group ring of $G$ over $\mathbb{Z}$** is the [[free abelian group]] $\mathbb{Z}[G]$, viewed as a $\mathbb{Z}$-[[algebra]] by [[bilinear map|bilinearly]] extending multiplication on $G$. > The **group algebra** or **group ring of $G$ over $R$** is [[tensor product of modules|the]] [[extension of scalars]] $R[G]:=\mathbb{Z}[G] \otimes _\mathbb{Z} R$, viewed as an $R$-[[algebra]] via the induced multiplication. > The [[category|category-theoretic]] perspective mimics that of the analogous situation in [[universal enveloping algebra]] for [[Lie algebra|Lie algebras]]. At some point I will explicitly lay that out here. ^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 > ```