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 : 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. 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: where 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. 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 .