---
tag: Y1P96JQN34
kind: definition
name: Group ring
msc: [16S34, 20C05]
defines: 4NMTSX0XDS
uses: [G5S1M9, 0B1261B5ED, 1HM7ZYMPD7, MNP2HBWB5Q, 0NNRDW10RZ]
sources:
  - url: https://en.wikipedia.org/wiki/Group_ring
    locator: Definition
---

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity and let $G$ be a [[group|G5S1M9]].
The *group ring* $R[G]$ is the set of functions $a \colon G \to R$ such that $a(g) \ne 0$ for only finitely many $g \in G$.
Write such a function as a finite formal sum $\sum_{g \in G} a_g \, g$ with $a_g = a(g)$.
Addition is termwise, and multiplication is
$$\Big(\sum_{g} a_g \, g\Big)\Big(\sum_{h} b_h \, h\Big) = \sum_{x \in G} \Big(\sum_{gh = x} a_g b_h\Big) x.$$

## Remarks

With these operations $R[G]$ is a ring with identity $1_R \, e$, where $e$ is the identity of $G$.
The maps $r \mapsto r e$ and $g \mapsto 1_R \, g$ embed $R$ as a subring and, when $R \ne 0$, $G$ into the group of units of $R[G]$.
If $R$ is commutative, $R[G]$ is an algebra over $R$, and if $R$ is a field, it is the [[group algebra|MNP2HBWB5Q]].

Example: let $G = \langle t \rangle$ be infinite cyclic.
A finite sum $\sum_n a_n t^n$ is a Laurent polynomial, and the product formula is the usual product of Laurent polynomials, so $R[G] \cong R[t, t^{-1}]$.

Non-example: for infinite $G$, the set of all functions $G \to R$ with the same product formula is not a ring in general.
For $G = \Z$ and $a = b = \sum_{n \in \Z} t^n$, the coefficient of $t^0$ in $ab$ would be the infinite sum $\sum_{n} 1$, which is not defined in $R = \Z$.

If $G$ has an element $g \ne e$ of finite order $n$, then $(1 - g)(1 + g + \cdots + g^{n-1}) = 1 - g^n = 0$, so $R[G]$ has [[zero divisors|0NNRDW10RZ]] when $R \ne 0$.
The group ring is the [[monoid ring|1HM7ZYMPD7]] of $G$ over $R$.