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  "body": "\n## Summary\n\nThe monoid ring $R[M]$ of a [[monoid|0N6R902FG1]] $M$ over a [[ring|0B1261B5ED]] $R$ is the ring of finite formal sums $\\sum r_m m$, multiplied with the monoid law.\nIt contains $R$ and a copy of $M$ inside its multiplicative monoid.\nPolynomial rings, Laurent polynomial rings and group rings are all monoid rings.\nIt is the free way to turn a monoid into a ring with coefficients in $R$.\n\n## Explanations\n\n### L1 human\n\nTake the elements of a monoid as building blocks and form linear combinations of them with coefficients in a ring.\nMultiply two combinations by expanding and multiplying the blocks with the monoid law.\nFor the monoid $\\set{1, x, x^2, \\dots}$ of powers of one letter, this gives the ordinary polynomials.\nFor example, $(1 + x)(2 + x^2) = 2 + 2x + x^2 + x^3$.\nChanging the monoid changes the ring: the integers under addition give polynomials with negative exponents too.\n",
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