Trivial extensions defined by Prüfer conditions

Abstract

Communicated by A.V. Geramita MSC: 13F05 13B05 13A15 13D05 13B25 a b s t r a c t This paper deals with well-known extensions of the Prüfer domain concept to arbitrary commutative rings. We investigate the transfer of these notions in trivial ring extensions (also called idealizations) of commutative rings by modules and then generate original families of rings with zero-divisors subject to various Prüfer conditions. The new examples give further evidence for the validity of the Bazzoni-Glaz conjecture on the weak global dimension of Gaussian rings. Moreover, trivial ring extensions allow us to widen the scope of validity of Kaplansky-Tsang conjecture on the content ideal of Gaussian polynomials.

Properties of chains of prime ideals in an amalgamated algebra along an ideal

“…Let K be a field and set A := K (3) , B := K (2) , and J := {0} × K , where K (n) is the direct product ring K × K × · · · × K (n-times). If f is the projection defined by (a, b, c) → (a, b), it is immediately seen that A f J ∼ = K (4) . Then Tot(A f J) ∼ = K (4) , but Tot(A) × Tot(B) ∼ = K (5) .…”

Prüfer Conditions in Commutative Rings

“…Recently, progress has been made in solving this conjecture under some restrictions on the ring R. Lucas (2008 [43]) provided an answer when the ring is reduced: Bakkari, Kabbaj and Mahdou (2010 [5]) investigated Kaplansky's Conjecture for trivial ring extensions. Let A be a ring and let E be an A module.…”

“…Bakkari, Kabbaj and Mahdou [5] called a ring over which every Gaussian polynomial has locally principal content ideal, a pseudo-arithmetical ring. It will be interesting to characterize pseudo-arithmetical rings, and also to clarify the situation for general rings, namely:…”

On Valuation Rings

“…In [13] C. U. Jensen gives some more characterization of arithmetical ring, and it is proved that a ring A is an arithmetical ring if and only if every localization A at a maximal (prime) ideal is a valuation ring. See for instance [1,2,5,6,8,9,13]. Let A be a ring, E be an A-module, and R = A ∝ E be the set of pairs a e with pairwise addition and multiplication given by a e b f = ab af + be .…”

“…These have proven to be useful in solving ON VALUATION RINGS 177 many open problems and conjectures for various contexts in (commutative and noncommutative) ring theory. See for instance [1,2,9,12,14,15,20].…”