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
f = f_0 + f_1 + \ldots + f_ n
is the decomposition of f into homogeneous pieces in R then f_ i \in I for each i.
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