group ring

For any group $G$ , the group ring $\mathbb{Z}[G]$ is defined to be the ring whose additive group is the abelian group of formal integer linear combinations of elements of $G$ , and whose multiplication operation is defined by multiplication in $G$ , extended $\mathbb{Z}$ –linearly to $\mathbb{Z}[G]$ .

More generally, for any ring $R$ , the group ring of $G$ over $R$ is the ring $R[G]$ whose additive group is the abelian group of formal $R$ –linear combinations of elements of $G$ , i.e.:

R[G]:=\left\{\left.\sum_{i=1}^{n}r_{i}g_{i}\ \right|\ r_{i}\in R,\ g_{i}\in G% \right\},

and whose multiplication operation is defined by $R$ –linearly extending the group multiplication operation of $G$ . In the case where $K$ is a field, the group ring $K[G]$ is usually called a group algebra.

Title	group ring
Canonical name	GroupRing
Date of creation	2013-03-22 12:13:27
Last modified on	2013-03-22 12:13:27
Owner	djao (24)
Last modified by	djao (24)
Numerical id	8
Author	djao (24)
Entry type	Definition
Classification	msc 20C05
Classification	msc 20C07
Classification	msc 16S34
Defines	group algebra