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On t- Spec(R[[X]])

Published online by Cambridge University Press:  20 November 2018

David E. Dobbs  and
Evan G. Houston
David E. Dobbs
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300 U.S.A.
Evan G. Houston
Affiliation:
Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223, U.S.A.
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Abstract

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Let D be an integral domain, and let X be an analytic indeterminate. As usual, if I is an ideal of D, set It = ∪{JV = (J-1)-1 | J is a nonzero finitely generated subideal of I}; this defines the t-operation, a particularly useful star-operation on D. We discuss the t-operation on R[[X]], paying particular attention to the relation between t- dim(R) and t- dim(R[[X]]). We show that if P is a t-prime of R, then P[[X]] contains a t-prime which contracts to P in R, and we note that this does not quite suffice to show that t- dim(R[[X]]) ≥ t- dim(R) in general. If R is Noetherian, it is easy to see that t- dim(R[[X]]) ≥ t- dim(R), and we show that we have equality in the case of t-dimension 1. We also observe that if V is a valuation domain, then t-dim(V[[X]]) ≥ t- dim(V), and we give examples to show that the inequality can be strict. Finally, we prove that if V is a finite-dimensional valuation domain with maximal ideal M, then MV[[X]] is a maximal t-ideal of V[[X]].

Keywords

13C1513F2513G0513F30
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 38 , Issue 2 , 01 June 1995 , pp. 187 - 195
Copyright
Copyright © Canadian Mathematical Society 1995

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