Given a commutative unit ring and a filtration
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of ideals of , the associated graded ring of with respect to is the graded ring
(2)
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The addition is defined componentwise, and the product is defined as follows. If is the residue class of mod , and is the residue class of mod , then is the residue class of mod .
is a quotient ring of the Rees ring of with respect to ,
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If is a proper ideal of , then the notation indicates the associated graded ring of with respect to the -adic filtration of ,
(4)
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If is Noetherian, then is as well. Moreover is finitely generated over . Finally, if is a local ring with maximal ideal , then
(5)
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