Homogeneous ideal

Remarks

Equivalent conditions: $I = \bigoplus_{n \ge 0} (I \cap R_n)$; and $I$ is generated by homogeneous elements. Example: in $k[x, y, z]$, with $k$ a field, the ideal $(x^2 - yz)$ is homogeneous, since it is generated by the homogeneous element $x^2 - yz$. Non-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 $(x - 1)$. If $I$ is homogeneous, then $R/I$ is graded by the images of the $R_n$. The ideal $R_+ = \bigoplus_{n \ge 1} R_n$ is homogeneous; it is called the irrelevant ideal.