Associated 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 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.
Generalization to multiplicative filtrations
The associated graded can also be defined more generally for multiplicative descending filtrations of R, 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.
- Eisenbud, Corollary 5.5