Prüfer Conditions in an Amalgamated Duplication of a Ring Along an Ideal

Abstract

ABSTRACT. This paper investigates ideal-theoretic as well as homological extensions of the Prüfer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prüfer conditions.

Bi-amalgamations subject to the arithmetical property

“…where w. dim(R) denotes the weak global dimension of R. Recall that all these properties are identical to the notion of Prüfer domain if R has no zero-divisors, and that the above implications are irreversible, in general, as shown by examples provided in [1,4,5,6,8,9,10,12,13,22,23,24]. Very recently, these conditions (among other Prüfer conditions) were thoroughly investigated in various contexts of duplications [12].…”

“…This construction was introduced in [26] as a natural generalization of duplications [12,14,17,18,27,30] and amalgamations [15,16,19]. In [26], the authors provide original examples of bi-amalgamations and, in particular, show that Boisen-Sheldon's CPI-extensions [7] can be viewed as bi-amalgamations (Notice that [15,Example 2.7] shows that CPI-extensions can be viewed as quotient rings of amalgamated algebras).…”

Prüfer conditions in amalgamated algebras

“…Assuming that J and f −1 (J) are regular ideals, Finocchiaro in [10, Theorem 3.1] obtained that R ⊲⊳ f J can be Prüfer ring only in the trivial case, namely when R and S are Prüfer rings and J = S. He also established some results about the transfer of some Prüfer-like conditions between R ⊲⊳ f J and R. Meanwhile, among other things, the authors in [4,Theorem 2.2] find necessary and sufficient conditions under which Prüfer-like properties transfer between the amalgamated duplication R ⊲⊳ I = R ⊲⊳ f I of a local ring R and the ring R itself, where I is an ideal of R, S = R and f is the identity map on R. In particular, they proved that R ⊲⊳ I is a Prüfer ring if and only if R is a Prüfer ring and I = rI for every regular element r of the unique maximal ideal of R. They then asked that is their characterization valid in the global case? i.e., when R is Prüfer (not necessarily local) or locally Prüfer [4,Question 2.11].…”

Gaussian and Prüfer conditions in bi-amalgamated algebras

“…They also established an analogue of Bazzoni-Glaz results on the transfer of Prüfer conditions between a ring and its total ring of quotients. In [9], the authors studied the transfer of the notions of local Prüfer ring and total ring of quotients. They examined the arithmetical, Gaussian, fqp conditions to amalgamated duplication along an ideal.…”