Almost perfect domains

Abstract

Abstract. Commutative rings all of whose quotients over non-zero ideals are perfect rings are called almost perfect. Revisiting a paper by J. R. Smith on local domains with TTN, some basic results on these domains and their modules are obtained. Various examples of local almost perfect domains with different features are exhibited. Introduction.A commutative ring R with 1 is called almost perfect if every quotient of R over a non-zero ideal is a perfect ring. In a recent paper [6] we characterized the commutative integral domains R which are almost perfect as those domains such that all R-modules have a strongly flat cover, or equivalently, such that all flat R-modules are strongly flat. The last property amounts to saying that weakly cotorsion modules (that is, cotorsion in Matlis' sense) are cotorsion (in Enochs' sense).The goal of this paper is to investigate more deeply almost perfect commutative rings. In the local case, almost perfect domains have already been introduced by J. R. Smith in [17] under the name of domains with TTN. We will reconsider here some of his results. We concentrate on almost perfect domains, because we show in Section 1 that an almost perfect ring that is not a domain is perfect.J. R. Smith [17] proved that if every torsion module over a domain R is semi-artininan, then R is locally almost perfect. An important property is missing in order to prove the converse, namely, h-locality (see [6]). We show that a classical example of almost Dedekind domain constructed by , which fails to be h-local, has all its torsion modules semi-artinian. If R is a local almost perfect domain and Q denotes its field of quotients, we prove that the Loewy length of Q/R equals ω if and only if the maximal ideal of R is almost nilpotent.In Section 3 we exhibit three different types of construction of almost perfect local domains. The first construction enables us to provide an example of an integrally closed local almost perfect domain which is not a

Loewy length of modules over almost perfect domains

“…In the local case, another characterization of almost perfect domains R in terms of their torsion modules is available, namely, every torsion R-module is semiartinian or, equivalently, the single module Q/R is semiartinian, where Q denotes the field of quotients of R. This result follows from the propositions at p. 239 of [7] and from the proof of Theorem 4.4.1 of [4] (see also [3] Corollary 2.4). Note that the module Q/R generates the class of the divisible R-modules, since almost perfect domains are Matlis domains (i.e., p.d.…”

“…We first recall the standard characterizing property of almost perfect local domains, which is an immediate consequence of the characterizations of perfect rings given by Bass [1, Theorem P]; see also [3,7,8]):…”

Weak-injectivity and almost perfect domains

“…Bazzoni and Salce [2] call a ring R almost perfect if all proper factor rings of R are perfect rings. Almost perfect domains can be characterized in several ways, one of which is that the concepts of Matlis-cotorsion and Enochs-cotorsion coincide (see also Göbel and Trlifaj [7,p.…”

Commutative ideal theory without finiteness conditions: Completely irreducible ideals

“…The almost perfect domains introduced by Bazzoni-Salce [4] are defined as domains such that every proper factor ring is a perfect ring. They prove that an almost perfect domain can be characterized as an h-local domain R such that for every nonzero proper ideal A, the factor ring R/A contains a simple R-module.…”

“…Our goal in this paper is to examine the structure of completely irreducible ideals of a commutative ring on which there are imposed no finiteness conditions. Other recent papers that address the structure and ideal theory of rings without finiteness conditions include [3], [4], [8], [10], [14], [15], [16], [19], [25], [26].…”