Graded ring

The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra.

Generally, the index set of a graded ring is supposed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article.

The corresponding idea in module theory is that of a graded module, namely a left module M over a graded ring R such that also

Example: a graded vector space is an example of a graded module over a field (with the field having trivial grading).

Example: a graded ring is a graded module over itself. An ideal in a graded ring is homogeneous if and only if it is a graded submodule. The annihilator of a graded module is a homogeneous ideal.

Remark: To give a graded morphism from a graded ring to another graded ring with the image lying in the center is the same as to give the structure of a graded algebra to the latter ring.

A graded module is said to be finitely generated if the underlying module is finitely generated. The generators may be taken to be homogeneous (by replacing the generators by their homogeneous parts.)

An algebra A over a ring R is a graded algebra if it is graded as a ring.

In the case where the ring R is also a graded ring, then one requires that

Graded algebras are much used in commutative algebra and algebraic geometry, homological algebra, and algebraic topology. One example is the close relationship between homogeneous polynomials and projective varieties (cf. homogeneous coordinate ring.)

The above definitions have been generalized to rings graded using any monoid G as an index set. A G-graded ring R is a ring with a direct sum decomposition

In formal language theory, given an alphabet A, the free monoid of words over A can be considered as a graded monoid, where the gradation of a word is its length.

Matsumura, H. (1989). "5 Dimension theory §S3 Graded rings, the Hilbert function and the Samuel function". . Cambridge Studies in Advanced Mathematics. 8. Translated by Reid, M. (2nd ed.). Cambridge University Press. ISBN 978-1-107-71712-1.