What is "group ring"

n. (context algebra English) Given ring ''R'' with identity not equal to zero, and group $G\; =\; \{g\_1,\; g\_2,\; ...,\; g\_n\}$, the group ring ''RG'' has elements of the form $a\_1\; g\_1\; +\; a\_2\; g\_2\; +\; ...\; +\; a\_n\; g\_n$ (where $a\_i\; isin\; R$) such that the sum of $a\_1\; g\_1\; +\; a\_2\; g\_2\; +\; ...\; +\; a\_n\; g\_n$ and $b\_1\; g\_1\; +\; b\_2\; g\_2\; +\; ...\; +\; b\_n\; g\_n$ is $(a\_1\; +\; b\_1)\; g\_1\; +\; (a\_2\; +\; b\_2)\; g\_2\; +\; ...\; +\; (a\_n\; +\; b\_n)\; g\_n$ and the product is $sum\_\{k=1\}^n\; left\; (\; sum\_\{g\_i\; g\_j\; =\; g\_k\}\; a\_i\; b\_j\; right\; )\; g\_k$.

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is one-to-one with the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.

If the given ring is commutative, a group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring.

The apparatus of group rings is especially useful in the theory of group representations.