The Rees ring of a commutative unit ring 
with respect to a filtration
 |
(1)
|
of ideals of 
is
 |
(2)
|
which is the set of all formal polynomials in the variable 
in which the coefficient of
lies in 
. 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]](/images/equations/ReesRing/Inline6.svg)
. It is also a subring of the
extended Rees ring
 |
(3)
|
which is a subring of ![R[t^(-1),t]](/images/equations/ReesRing/Inline7.svg)
, the ring of all finite
linear combinations of integer
(possibly negative) powers of 
.
If 
is a proper
ideal of 
,
the notation 
(or 
) indicates the (extended) Rees ring
of 
with respect to the 
-adic filtration of 
. If 
is the polynomial ring ![K[X_1,...,X_n]](/images/equations/ReesRing/Inline17.svg)
over a field 
, then the algebraic blow-up
of the affine space 
along the closed subscheme
defined by 
is 
, formed from the nonnegative
graded ring 
(Stacks Project 2026).
