Graded ring

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This article defines a notion of a ring with additional structure


A graded ring is a commutative unital ring A equipped with a direct sum decomposition as a sum of Abelian subgroups:

A = \oplus_{i=-\infty}^\infty A_i = \cdots A_{-2} \oplus A_{-1} \oplus A_0 \oplus A_1 \oplus A_2 \oplus \cdots

such that the following hold:

  • Each A_i is a subgroup under addition
  • 1 \in A_0
  • A_mA_n \subset A_{m+n}. In other words, if a \in A_m and b \in A_n then ab \in A_{m+n}

A structure of the above sort on a ring is termed a gradation, also a \mathbb{Z}-gradation. The ring A is positively graded if A_i = 0 for all i<0.

There are related notions for noncommutative rings.

Related notions

Weaker notions