Given a commutative unit ring and a filtration
(1)
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of ideals of , the Rees ring of with respect to is
(2)
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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 . It is also a subring of the extended Rees ring
(3)
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which is a subring of , 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 over a field , then is the coordinate ring of the blow-up of the affine space along the affine variety .