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    On prime ideals of noetherian skew power series rings

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    Genre
    Journal Article
    Date
    2012-12-01
    Author
    Letzter, ES
    Subject
    math.RA
    math.RA
    16W60; 16W80; 16S36; 16P40; 11R23; 16W35
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/6005
    
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    DOI
    10.1007/s11856-012-0048-6
    Abstract
    We study prime ideals in skew power series rings T:= R[[y; τ, δ]], for suitably conditioned complete right Noetherian rings R, automorphisms τ of R, and τ-derivations δ of R. Such rings were introduced by Venjakob, motivated by issues in noncommutative Iwasawa theory. Our main results concern "Cutting Down" and "Lying Over." In particular, assuming that τ extends to a compatible automorphsim of T, we prove: If I is an ideal of R, then there exists a τ-prime ideal P of T contracting to I if and only if I is a τ-δ-prime ideal of R. Consequently, under the more specialized assumption that δ = τ - id (a basic feature of the Iwasawa-theoretic context), we can conclude: If I is an ideal of R, then there exists a prime ideal P of T contracting to I if and only if I is a τ-prime ideal of R. Our approach depends essentially on two key ingredients: First, the algebras considered are Zariskian (in the sense of Li and Van Oystaeyen), and so the ideals are all topologically closed. Second, topological arguments can be used to apply previous results of Goodearl and the author on skew polynomial rings. © 2012 Hebrew University Magnes Press.
    Citation to related work
    Springer Science and Business Media LLC
    Has part
    Israel Journal of Mathematics
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    ae974a485f413a2113503eed53cd6c53
    http://dx.doi.org/10.34944/dspace/5987
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