# 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]

where

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

## Examples

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

## References

- Eisenbud, Corollary 5.5
- 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