Graded Algebra - Graded Ring

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+...+a_n with all a_i homogeneous elements, then all the a_i 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 A_i = 0 for i > 0. This is called the trivial gradation on A.