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