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An intermediate ring between a polynomial ring and a power series ring

M. Tamer Koşan, Tsiu-Kwen Lee, Yiqiang Zhou (2013)

Colloquium Mathematicae

Let R[x] and R[[x]] respectively denote the ring of polynomials and the ring of power series in one indeterminate x over a ring R. For an ideal I of R, denote by [R;I][x] the following subring of R[[x]]: [R;I][x]: = ∑ i ≥ 0 r i x i ∈ R [ [ x ] ] : ∃ 0 ≤ n∈ ℤ such that r i ∈ I , ∀ i ≥ n. The polynomial and power series rings over R are extreme cases where I = 0 or R, but there are ideals I such that neither R[x] nor R[[x]] is isomorphic to [R;I][x]. The results characterizing polynomial rings or power series rings with a certain ring...

Dimension in algebraic frames, II: Applications to frames of ideals in C ( X )

Jorge Martinez, Eric R. Zenk (2005)

Commentationes Mathematicae Universitatis Carolinae

This paper continues the investigation into Krull-style dimensions in algebraic frames. Let L be an algebraic frame. dim ( L ) is the supremum of the lengths k of sequences p 0 < p 1 < ⋯ < p k of (proper) prime elements of L . Recently, Th. Coquand, H. Lombardi and M.-F. Roy have formulated a characterization which describes the dimension of L in terms of the dimensions of certain boundary quotients of L . This paper gives a purely frame-theoretic proof of this result, at once generalizing it to frames which are not necessarily...

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