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).
