Nebraska Cooperative Fish & Wildlife Research Unit


Date of this Version



Michigan Math. J. 57 (2008)


W. Heinzer, C. Rotthaus, & S. Wiegand


Let R be a commutative ring with identity. A filtration on R is a decreasing sequence {In}∞ n=0 of ideals of R. Associated to a filtration is a well-defined completion R ∗ = l←i−mn R/In and a canonical homomorphism ψ : R → R ∗ [13, Chap. 9]. If ∞ n=0 In = (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 {In}∞ n=0 is multiplicative if I0 = R and InIm ⊆ In+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.