Rees Module
Given a module M over 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 module of M with respect to F is
| R_+(F,M)= direct sum _(i=0)^inftyI_iMt^i, |
(2)
|
which is the set of all formal polynomials in the variable t in which the coefficient of t^i is of the form am, where a in I_i and m in M. It is a graded module over the Rees ring R_+(F).
The subscript + distinguishes it from the so-called extended Rees module, defined as
| R(F,M)= direct sum _(i=-infty)^inftyI_iMt^i, |
(3)
|
where R(F,M)=R for all i<0. This module includes all polynomials containing negative powers of t.
If I is a proper ideal of R, the notation R_+(I,M) (or R(I,M)) indicates the (extended) Rees module of M with respect to the I-adic filtration.
See also
Associated Graded Module, Rees RingThis entry contributed by Margherita Barile
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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 ModuleCite this as:
Barile, Margherita. "Rees Module." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ReesModule.html