On maximal ideals of the polynomial ring and some conjectures on Ext-index of rings

Abstract

Recall that a ring [Formula: see text] is a Hilbert ring if any maximal ideal of [Formula: see text] contracts to a maximal ideal of [Formula: see text]. The main purpose of this paper is to characterize the prime ideals of a commutative ring [Formula: see text] which are traces of the maximal ideals of the polynomial ring [Formula: see text]. In this context, we prove that if [Formula: see text] is a prime ideal of [Formula: see text] such that [Formula: see text] is a semi-local domain of (Krull) dimension [Formula: see text], then [Formula: see text] is the trace of a maximal ideal of [Formula: see text]. Whereas, if [Formula: see text] is Noetherian and either ([Formula: see text]) or (the quotient field of [Formula: see text] is algebraically closed, [Formula: see text] and [Formula: see text] is not semi-local), then [Formula: see text] is never the trace of a maximal ideal of [Formula: see text]. Putting these results into use in investigating the Ext-index of Noetherian rings, we establish connections between the finiteness of the Ext-index of localizations of the polynomial rings [Formula: see text] and the finiteness of the Ext-index of localizations of [Formula: see text]. This allows us to provide a new class of rings satisfying some known conjectures on Ext-index of Noetherian rings as well as to build bridges between these conjectures.