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  "body": "\n## Summary\n\nThe group ring $R[G]$ of a [[group|G5S1M9]] $G$ over a [[ring|0B1261B5ED]] $R$ is the ring of finite formal sums $\\sum_g a_g g$ with coefficients in $R$.\nSuch sums are multiplied using the group law.\nIt turns questions about $G$ and its actions into questions about modules over one ring.\nWhen $R$ is a field, it is the [[group algebra|MNP2HBWB5Q]], the basic object of representation theory.\nIt is the special case of the [[monoid ring|1HM7ZYMPD7]] in which the monoid is a group.\n\n## Explanations\n\n### L1 human\n\nThink of the elements of $G$ as basis vectors, and allow coefficients from $R$.\nMultiply two formal sums by expanding and multiplying group elements in $G$.\nFor the integers $\\Z$ and the cyclic group $\\set{1, t}$ of order $2$, the element $1 - t$ satisfies $(1 - t)^2 = 1 - 2t + t^2 = 2 - 2t = 2(1 - t)$.\nThe integral group ring $\\Z[G]$ is important in topology and in the study of group extensions.\n",
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