Homogenous / Graded Ideal

Let $A=\bigoplus_{i=0}^\infty A_i$ be a graded ring. An ideal $I\subset A$ is homogenous (also called graded) if for every element $x\in I$ , its homogenous components also belong to $I$ .

An ideal in a graded ring is homogenous if and only if it is a graded submodule. The intersections of a homogenous ideal $I$ with the $A_i$ are called the homogenous parts of $I$ . A homogenous ideal $I$ is the direct sum of its homogenous parts, that is, $\displaystyle I=\bigoplus_{i=0}^\infty (I\cap A_i).$