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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity and let $M$ be a [[monoid|0N6R902FG1]], written multiplicatively with identity $e$.\nThe *monoid ring* $R[M]$ is the set of functions $f \\colon M \\to R$ with $f(m) \\ne 0$ for only finitely many $m \\in M$.\nSuch a function is written as a finite formal sum $\\sum_{m \\in M} f(m) \\, m$.\nAddition is pointwise, and multiplication is the convolution\n$$(f g)(m) = \\sum_{\\substack{x, y \\in M \\\\ x y = m}} f(x) \\, g(y).$$\nThe sum is finite because $f$ and $g$ have finite support.\nWith these operations $R[M]$ is a ring with identity $1 \\cdot e$.\n\n## Remarks\n\nExample: for the monoid $M = \\set{1, x, x^2, \\dots}$ with $x^i x^j = x^{i+j}$, the convolution is the usual product of polynomials, so $R[M]$ is the [[polynomial ring|0SDC934R52]] $R[x]$.\nFor instance $(1 + x)(1 - x) = 1 \\cdot 1 + (1 - 1) x - x^2 = 1 - x^2$.\nFor a [[group|G5S1M9]] $G$, $R[G]$ is the [[group ring|4NMTSX0XDS]], and for $M = (\\Z, +)$ it is the [[Laurent polynomial ring|QMY5JTZZB8]].\nNon-example: the set of all functions $\\Z \\to R$ with the same product is not a ring, because the finiteness condition fails and the convolution sum $\\sum_{i + j = n} f(i) g(j)$ can be an infinite sum.\nThe ring $R[M]$ is commutative when $R$ and $M$ are commutative.\nWhen $R$ is commutative, each monoid homomorphism from $M$ to the multiplicative monoid of an $R$-algebra $A$ extends uniquely to an $R$-algebra homomorphism $R[M] \\to A$.\nIn additive notation for $M$, the same ring is often called the monoid algebra.\n",
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