A graded ring A is a ring that has a direct sum decomposition into (abelian) additive groups
such that the ring multiplication satisfies
and so
Elements of any factor of the decomposition are known as homogeneous elements of degree n. An ideal or other subset ⊂ A is homogeneous if every element a ∈ is the sum of homogeneous elements that belong to For a given a these homogeneous elements are uniquely defined and are called the homogeneous parts of a. Equivalently, an ideal is homogeneous if for each a in the ideal, when a=a1+a2+...+an with all ai homogeneous elements, then all the ai are in the ideal.
If I is a homogeneous ideal in A, then is also a graded ring, and has decomposition
Any (non-graded) ring A can be given a gradation by letting A0 = A, and Ai = 0 for i > 0. This is called the trivial gradation on A.
Read more about this topic: Graded Algebra
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