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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity and let $G$ be a [[group|G5S1M9]].\nThe *group ring* $R[G]$ is the set of functions $a \\colon G \\to R$ such that $a(g) \\ne 0$ for only finitely many $g \\in G$.\nWrite such a function as a finite formal sum $\\sum_{g \\in G} a_g \\, g$ with $a_g = a(g)$.\nAddition is termwise, and multiplication is\n$$\\Big(\\sum_{g} a_g \\, g\\Big)\\Big(\\sum_{h} b_h \\, h\\Big) = \\sum_{x \\in G} \\Big(\\sum_{gh = x} a_g b_h\\Big) x.$$\n\n## Remarks\n\nWith these operations $R[G]$ is a ring with identity $1_R \\, e$, where $e$ is the identity of $G$.\nThe 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 $R[G]$.\nIf $R$ is commutative, $R[G]$ is an algebra over $R$, and if $R$ is a field, it is the [[group algebra|MNP2HBWB5Q]].\n\nExample: let $G = \\langle t \\rangle$ be infinite cyclic.\nA finite sum $\\sum_n a_n t^n$ is a Laurent polynomial, and the product formula is the usual product of Laurent polynomials, so $R[G] \\cong R[t, t^{-1}]$.\n\nNon-example: for infinite $G$, the set of all functions $G \\to R$ with the same product formula is not a ring in general.\nFor $G = \\Z$ 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 $R = \\Z$.\n\nIf $G$ 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 $R \\ne 0$.\nThe group ring is the [[monoid ring|1HM7ZYMPD7]] of $G$ over $R$.\n",
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