In mathematics, the '''associated graded ring''' of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring: :<math>\operatorname{gr}_I R = \bigoplus_{n=0}^\infty I^n/I^{n+1}</math>. Similarly, if ''M'' is a left ''R''-module, then the '''associated graded module''' is the graded module over <math>\operatorname{gr}_I R</math>: :<math>\operatorname{gr}_I M = \bigoplus_{n=0}^\infty I^n M/ I^{n+1} M</math>.
== Basic definitions and properties == For a ring ''R'' and ideal ''I'', multiplication in <math>\operatorname{gr}_IR</math> is defined as follows: First, consider homogeneous elements <math>a \in I^i/I^{i + 1}</math> and <math>b \in I^j/I^{j + 1}</math> and suppose <math>a' \in I^i</math> is a representative of ''a'' and <math>b' \in I^j</math> is a representative of ''b''. Then define <math>ab</math> to be the equivalence class of <math>a'b'</math> in <math>I^{i + j}/I^{i + j + 1}</math>. Note that this is well-defined modulo <math>I^{i + j + 1}</math>. 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 <math>f \in M</math>, the '''initial form''' of ''f'' in <math>\operatorname{gr}_I M</math>, written <math>\mathrm{in}(f)</math>, is the equivalence class of ''f'' in <math>I^mM/I^{m+1}M</math> where ''m'' is the maximum integer such that <math>f\in I^mM</math>. If <math>f \in I^mM</math> for every ''m'', then set <math>\mathrm{in}(f) = 0</math>. The initial form map is only a map of sets and generally not a homomorphism. For a submodule <math>N \subset M</math>, <math>\mathrm{in}(N)</math> is defined to be the submodule of <math>\operatorname{gr}_I M</math> generated by <math>\{\mathrm{in}(f) | f \in N\}</math>. This may not be the same as the submodule of <math>\operatorname{gr}_IM</math> 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 <math>\operatorname{gr}_I R</math> is an integral domain, then ''R'' is itself an integral domain.<ref>{{harvnb|Eisenbud|1995|loc=Corollary 5.5}}</ref>
== gr of a quotient module == Let <math>N \subset M</math> be left modules over a ring ''R'' and ''I'' an ideal of ''R''. Since :<math>{I^n(M/N) \over I^{n+1}(M/N)} \simeq {I^n M + N \over I^{n+1}M + N} \simeq {I^n M \over I^n M \cap (I^{n+1} M + N)} = {I^n M \over I^n M \cap N + I^{n+1} M}</math> (the last equality is by modular law), there is a canonical identification:<ref>{{harvnb|Zariski|Samuel|1975|loc=Ch. VIII, a paragraph after Theorem 1.}}</ref> :<math>\operatorname{gr}_I (M/N) = \operatorname{gr}_I M / \operatorname{in}(N)</math> where :<math>\operatorname{in}(N) = \bigoplus_{n=0}^{\infty} {I^n M \cap N + I^{n+1} M \over I^{n+1} M},</math> called the ''submodule'' generated by the initial forms of the elements of <math>N</math>.
== Examples ==
Let ''U'' be the universal enveloping algebra of a Lie algebra <math>\mathfrak{g}</math> over a field ''k''; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that <math>\operatorname{gr} U</math> is a polynomial ring; in fact, it is the coordinate ring <math>k[\mathfrak{g}^*]</math>.
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 :<math>R = I_0 \supset I_1 \supset I_2 \supset \dotsb</math> such that <math>I_jI_k \subset I_{j + k}</math>. The graded ring associated with this filtration is <math>\operatorname{gr}_F R = \bigoplus_{n=0}^\infty I_n/ I_{n+1}</math>. Multiplication and the initial form map are defined as above.
== See also == * Graded (mathematics) * Rees algebra
==References== {{reflist}} * {{cite book|last=Eisenbud|first=David|authorlink=David Eisenbud|title=Commutative Algebra|series=Graduate Texts in Mathematics|volume=150|publisher=Springer-Verlag|year=1995|isbn=0-387-94268-8|doi=10.1007/978-1-4612-5350-1|mr=1322960|location=New York}} * {{cite book|last=Matsumura|first=Hideyuki|title=Commutative ring theory|others=Translated from the Japanese by M. Reid|edition=Second|series=Cambridge Studies in Advanced Mathematics|volume=8|publisher=Cambridge University Press|location=Cambridge|year=1989|isbn=0-521-36764-6|mr=1011461}} * {{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative algebra. Vol. II | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90171-8 |mr=0389876 | year=1975}}
Category:Ring theory