Let R be a commutative ring with identity. A filtration on R is a decreasing sequence {I_n}_(n=0)~∞ of ideals of R. Associated to a filtration is a well-defined completion R~* = lim_n R/I_n and a canonical homomorphism ψ: R → R~* [13, Chap. 9]. If ∩_(n=0)~∞ I_n = (0), then ψ is injective and R may be regarded as a subring of R~* [13, p. 401]. In the terminology of Northcott, a filtration {I_n}_(n=0)~∞ is multiplicative if I_0 = R and I_nI_m is contained in I_(n+m) for all m ≥ 0 and n ≥ 0 [13, p. 408]. A well-known example of a multiplicative filtration on R is the I-adic filtration {I~n}_(n=0)~∞, where I is a fixed ideal of R.
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