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  "body": "\n## Definition\n\nLet $R = \\bigoplus_{n \\ge 0} R_n$ be a [[graded ring|8KB4SQGC0C]].\nAn [[ideal|096Z9MP2ME]] $I$ of $R$ is *homogeneous* if for every $f \\in I$ with homogeneous components $f_n \\in R_n$, each $f_n$ lies in $I$.\n\n## Remarks\n\nEquivalent conditions: $I = \\bigoplus_{n \\ge 0} (I \\cap R_n)$; and $I$ is [[generated|0GSW76Q14N]] by homogeneous elements.\nExample: in $k[x, y, z]$, with $k$ a field, the ideal $(x^2 - yz)$ is homogeneous, since it is generated by the homogeneous element $x^2 - yz$.\nNon-example: in $k[x]$ the ideal $(x - 1)$ is not homogeneous; it contains $x - 1$, whose component $-1$ of degree $0$ is a unit and does not lie in the proper ideal $(x - 1)$.\nIf $I$ is homogeneous, then $R/I$ is graded by the images of the $R_n$.\nThe ideal $R_+ = \\bigoplus_{n \\ge 1} R_n$ is homogeneous; it is called the irrelevant ideal.\n",
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