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).

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