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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]].\nA *grading* of $R$ is a family $(R_n)_{n \\ge 0}$ of subgroups of the [[abelian group|082GHNPJ53]] $(R, +)$ such that:\n\n1. $R = \\bigoplus_{n \\ge 0} R_n$ as abelian groups, that is, every $f \\in R$ is uniquely a finite sum $f = \\sum_n f_n$ with $f_n \\in R_n$;\n2. $R_m R_n \\subseteq R_{m+n}$ for all $m, n \\ge 0$.\n\nA *graded ring* is a ring with a grading.\nThe elements of $R_n$ are the homogeneous elements of degree $n$, and $f_n$ is the homogeneous component of $f$ of degree $n$.\n\n## Remarks\n\nExample: $R = k[x_1, \\dots, x_r]$, with $k$ a field and $R_n$ the span of the monomials of degree $n$; the product of two monomials of degrees $m$ and $n$ has degree $m + n$.\nNon-example: in $R = k[x]/(x^2 - 1)$ take $R_0 = k$, $R_1 = k \\bar x$ and $R_n = 0$ for $n \\ge 2$; clause 1 holds, but $\\bar x \\cdot \\bar x = 1$ is a nonzero element of $R_0$, not of $R_2 = 0$, so clause 2 fails.\nIt follows from the definition that $1 \\in R_0$ and that $R_0$ is a [[subring|0GNCRPYXYV]].\nOne also grades by $\\Z$ or by another monoid in the same way; the ring $R$ is then $\\Z$-graded, and so on.\n",
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