Homogeneous ideal

Ideals

An ideal${\displaystyle I\subseteq R}$ is homogeneous, if for every ⁠${\displaystyle a\in I}$⁠, the homogeneous components of ${\displaystyle a}$ also belong to ⁠${\displaystyle I}$⁠. (Equivalently, if it is a graded submodule of ⁠${\displaystyle R}$⁠; see § Graded module.) The intersection of a homogeneous ideal ${\displaystyle I}$ with ${\displaystyle R_{n}}$ is an ⁠${\displaystyle R_{0}}$⁠-submodule of ${\displaystyle R_{n}}$ called the homogeneous part of degree ${\displaystyle n}$ of ⁠${\displaystyle I}$⁠. A homogeneous ideal is the direct sum of its homogeneous parts.