Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T22:58:25.925Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Arnold, J. T., Krull dimension in power series rings, Trans. Amer. Math. Soc. 177(1973), 299—304.Google Scholar
2. Arnold, J. T., Power series rings over Prüfer domains, Pacific J. Math. 44(1973), 1—11.Google Scholar
3. Arnold, J. T., Power series rings over discrete valuation rings, Pacific J. Math. 93(1981), 31—33.Google Scholar
4. Arnold, J. T. and Brewer, J. W., On when (D[[X]])p[[X]] is a valuation domain, Proc. Amer. Math. Soc 37(1973), 326–332.Google Scholar
5. Anderson, D. F. and Kang, B. G., On Claborn's question, manuscript.Google Scholar
6. Brewer, J. W., Power Series Rings over Commutative Rings, Lecture Notes in Pure and Appl. Math. 64, Marcel Dekker, New York, 1981.Google Scholar
7. Bastida, E. and Gilmer, R., Overrings and divisorial ideals of rings of the form D + M, Michigan Math. J. 20(1973), 79–95.Google Scholar
8. Dobbs, D. E., Divided rings and going down, Pacific J. Math. 67(1976), 53—63.Google Scholar
9. Fields, D. E., Dimension theory in power series rings, Pacific J. Math. 35(1970), 601—611.Google Scholar
10. Gilmer, R., Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.Google Scholar
11. Houston, E. G., Prime t-ideals in R[X], Commutative ring theory (Proc. 1992 Fès Internat. Conf.), Lecture Notes in Pure and Appl. Math. 153, Marcel Dekker, New York, 1994.Google Scholar
12. Houston, E. G. and Hedstrom, J. R. Pseudo-valuation domains, Pacific J. Math. 75(1978), 137–147.Google Scholar
13. Jaffard, P., Les Systèmes d'Idéaux, Dunod, Paris, 1960.Google Scholar
14. Lequain, Y., Catenarian property in a domain of formal power series, J. Algebra 65( 1980), 110–117.Google Scholar
15. Maths, E., Torsion-Free Modules, Chicago Lectures in Math., The University of Chicago Press, Chicago, 1972.Google Scholar
16. Matsumura, H., Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.Google Scholar
17. Zariski, O. and Samuel, P., Commutative Algebra, vol. II, The University Series in Higher Math., Van Nostrand, Princeton, New Jersey, 1960.Google Scholar