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

Dobbs, David E. ; Houston, Evan G.

Canadian mathematical bulletin, Tome 38 (1995) no. 2, pp. 187-195

Résumé

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 ofR, 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]].

Détail

DOI : 10.4153/CMB-1995-027-1

Mots-clés : 13C15, 13F25, 13G05, 13F30