TOPICS
Search

Rees Ring


Given a commutative unit ring R and a filtration

 F:... subset= I_2 subset= I_1 subset= I_0=R
(1)

of ideals of R, the Rees ring of R with respect to F is

 R_+(F)=I_0 direct sum I_1t direct sum I_2t^2 direct sum ...,
(2)

which is the set of all formal polynomials in the variable t in which the coefficient of t^i lies in I_i. It is a graded ring with respect to the usual addition and multiplication of polynomials, which makes it a subring of the polynomial ring R[t]. It is also a subring of the extended Rees ring

 R(F)=...Rt^(-2) direct sum Rt^(-1) direct sum R direct sum I_0 direct sum I_1t direct sum I_2t^2 direct sum ...,
(3)

which is a subring of R[t^(-1),t], the ring of all finite linear combinations of integer (possibly negative) powers of t.

If I is a proper ideal of R, the notation R_+(I) (or R(I)) indicates the (extended) Rees ring of R with respect to the I-adic filtration of R. If R is the polynomial ring K[X_1,...,X_n] over a field K, then R(I) is the coordinate ring of the blow-up of the affine space K^n along the affine variety V(I).


See also

Associated Graded Ring, Rees Module

This entry contributed by Margherita Barile

Explore with Wolfram|Alpha

References

Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1993.Matsumura, H. Commutative Ring Theory. Cambridge, England: Cambridge University Press, 1986.

Referenced on Wolfram|Alpha

Rees Ring

Cite this as:

Barile, Margherita. "Rees Ring." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ReesRing.html

Subject classifications