Given a module over a commutative unit ring and a filtration
(1)
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of ideals of , the associated graded module of with respect to is
(2)
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which is a graded module over the associated graded ring with respect to the addition and the multiplication by scalars defined componentwise.
If is a proper ideal of , then the notation indicates the associated graded module of with respect to the -adic filtration of ,
(3)
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