Associated graded ring

In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring:


Similarly, if M is a left R-module, then the associated graded module is the graded module over :


Basic definitions and properties

For a ring R and ideal I, multiplication in is defined as follows: First, consider homogeneous elements and and suppose is a representative of a and is a representative of b. Then define to be the equivalence class of in . Note that this is well-defined modulo . Multiplication of inhomogeneous elements is defined by using the distributive property.

A ring or module may be related to its associated graded ring or module through the initial form map. Let M be an R-module and I an ideal of R. Given , the initial form of f in , written , is the equivalence class of f in where m is the maximum integer such that . If for every m, then set . The initial form map is only a map of sets and generally not a homomorphism. For a submodule , is defined to be the submodule of generated by . This may not be the same as the submodule of generated by the only initial forms of the generators of N.

A ring inherits some "good" properties from its associated graded ring. For example, if R is a noetherian local ring, and is an integral domain, then R is itself an integral domain.[1]

gr of a quotient module

Let be left modules over a ring R and I an ideal of R. Since

(the last equality is by modular law), there is a canonical identification:[2]


called the submodule generated by the initial forms of the elements of .


Let U be the universal enveloping algebra of a Lie algebra over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that is a polynomial ring; in fact, it is the coordinate ring .

The associated graded algebra of a Clifford algebra is an exterior algebra; i.e., a Clifford algebra degenerates to an exterior algebra.

Generalization to multiplicative filtrations

The associated graded can also be defined more generally for multiplicative descending filtrations of R (see also filtered ring.) Let F be a descending chain of ideals of the form

such that . The graded ring associated with this filtration is . Multiplication and the initial form map are defined as above.

See also


  1. Eisenbud, Corollary 5.5
  2. Zariski–Samuel, Ch. VIII, a paragraph after Theorem 1.
  • Eisenbud, David (1995). Commutative Algebra. Graduate Texts in Mathematics. 150. New York: Springer-Verlag. doi:10.1007/978-1-4612-5350-1. ISBN 0-387-94268-8. MR 1322960.
  • Matsumura, Hideyuki (1989). Commutative ring theory. Cambridge Studies in Advanced Mathematics. 8. Translated from the Japanese by M. Reid (Second ed.). Cambridge: Cambridge University Press. ISBN 0-521-36764-6. MR 1011461.
  • Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876
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