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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity and let $G$ be a [[group|G5S1M9]].\nLet $\\alpha \\colon G \\to \\Aut(R)$ be a group homomorphism, written $g \\mapsto \\alpha_g$; it is an [[action|0WCY42EV9V]] of $G$ on $R$ by ring automorphisms.\nThe *skew group ring* $R * G$, also written $R \\rtimes_\\alpha G$, is the set of finite formal sums $\\sum_{g \\in G} r_g g$ with $r_g \\in R$.\nAddition is termwise, and multiplication is the biadditive extension of\n$$(r g)(s h) = r \\, \\alpha_g(s) \\, gh \\qquad (r, s \\in R, \\ g, h \\in G).$$\n\n## Remarks\n\nThe product is associative because $\\alpha_{gh} = \\alpha_g \\circ \\alpha_h$: both $((rg)(sh))(tk)$ and $(rg)((sh)(tk))$ equal $r \\alpha_g(s) \\alpha_{gh}(t) \\, ghk$.\nThe identity is $1 \\cdot e$, where $e$ is the identity of $G$, and $R$ is a subring through $r \\mapsto r e$.\nIf $\\alpha$ is trivial, $R * G$ is the [[group ring|4NMTSX0XDS]] $R[G]$.\nExample: let $G = \\set{e, c}$ act on $\\C$ by complex conjugation, $\\alpha_c(z) = \\bar z$.\nThen $R * G = \\C * G$ has dimension $4$ over $\\R$.\nThe map sending $z$ to multiplication by $z$ and $c$ to conjugation, both as $\\R$-linear maps of $\\C = \\R^2$, is an isomorphism onto the [[matrix ring|0FHVPRAE60]] $M_2(\\R)$.\nNon-example: if $\\alpha \\colon G \\to \\Aut(R)$ is a map that is not a homomorphism, the same formula fails associativity, since $(1g)((1h)(s e)) = \\alpha_g(\\alpha_h(s)) gh$ while $((1g)(1h))(s e) = \\alpha_{gh}(s) gh$.\nWhen a twisting cocycle is also allowed, $g h$ is replaced by a scalar multiple of $gh$, and one gets a crossed product.\n",
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