Definition 111.27.1. Let $R$ be a graded ring. A *homogeneous* ideal is simply an ideal $I \subset R$ which is also a graded submodule of $R$. Equivalently, it is an ideal generated by homogeneous elements. Equivalently, if $f \in I$ and

is the decomposition of $f$ into homogeneous pieces in $R$ then $f_ i \in I$ for each $i$.

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