The weak dimensions of Gaussian rings

Abstract

Abstract. We provide necessary and sufficient conditions for a Gaussian ring R to be semihereditary, or more generally, of w.dimR ≤ 1. Investigating the weak global dimension of a Gaussian coherent ring R, we show that the only values w.dimR may take are 0, 1 and ∞; but that f P.dimR is always at most one. In particular, we conclude that a Gaussian coherent ring R is either Von Neumann regular, or semihereditary, or non-regular of w.dimR = ∞.

Open Problems in Commutative Ring Theory

“…If R is a Gaussian ring, then is w.dim R = 0; 1, or 1? This is the case for coherent Gaussian rings [67,Theorem 3.3] (and actually, more generally, for coherent Prüfer rings [14, Proposition 6.1]), arithmetical rings [97 and 14, remark in the last paragraph], and a particular case of Gaussian rings [14,Theorem 6.4]. [68,Corollary 6.7], in which case w.dim R = 0 or 1 ; and also holds for one example of a non coherent ring [68,Example 6.8].…”

“…Glaz [67] and Bazzoni & Glaz [14] consider, among other properties, the …nitistic and weak global dimensions of rings satisfying various Prüfer conditions.…”

“…Using the results obtained for the …nitistic projective dimension of rings (see previous problem) with various Prüfer conditions, it is possible to determine the values of the weak global dimensions of rings under certain Prüfer conditions [14,67], but many questions are not yet answered: Problem 2a. If R is a Gaussian ring, then is w.dim R = 0; 1, or 1?…”

Bazzoni–Glaz conjecture

“…We note that it follows from Osofsky [17] that arithmetical rings have weak global dimension at most one or ∞. In the same vein, the author of [7] is concerned with giving a homological characterization to Gaussian rings. She considers the question: For a Gaussian ring R, what possible values may w. gl.…”

Prüfer-Like Conditions in Subring Retracts and Applications

“…From Bazzoni and Glaz [4, Theorem 3.12], we note that a Prüfer ring R satisfies one of the five conditions if and only if the total ring of quotients T ot(R) of R satisfies the same condition. See for instance [3,4,7,10,11,13,17,23,28].…”