Given a commutative unit ring 
and a filtration
 |
(1)
|
of ideals of 
, the Rees ring of 
with respect to 
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/Inline8.svg)
.
It is also a subring of the extended Rees ring
 |
(3)
|
which is a subring of ![R[t^(-1),t]](/images/equations/ReesRing/Inline9.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/Inline19.svg)
over a field 
, then 
is the coordinate ring of the blow-up of the affine
space 
along the affine variety 
.
