Prüfer conditions in amalgamated algebras

Abstract

In this paper we improve the recent results on the transfer of Prüfer, Gaussian and arithmetical conditions on amalgamated constructions. As an application we provide an answer to a question posed by Chhiti, Jarrar, Kabbaj and Mahdou as well as we construct various examples.2010 Mathematics Subject Classification. Primary 13F05, 13A15.Proof. Assume that (r, f (r) + j) ∈ Z(R ⊲⊳ f J). Then (r, f (r) + j)(s, f (s) + j ′ ) = 0 for some non-zero element (s, f (s) + j ′ ) ∈ R ⊲⊳ f J. Hence rs = 0 and jf (s) + j ′ (f (r) + j) = 0. If s = 0, then j ′ = 0 and j ′ (f (r) + j) = 0. Otherwise, r ∈ Z(R). This proves the inclusion.We already have the inclusion {(r, f (r) + j) | j ′ (f (r) + j) = 0, for some j ′ ∈ J \ {0}} ⊆ Z(R ⊲⊳ f J). To complete the proof, it is enough for us to show that {(r, f (r) + j) | r ∈ Z(R) and j ∈ J} ⊆ Z(R ⊲⊳ f J) under the validity of any of

Hilbert rings with maximal ideals of different heights and unruly Hilbert rings

“…Amalgamation, in turn, can be realized as a pullback. The construction has proved its worth providing numerous examples and counterexamples in commutative ring theory ( [11], [20], [6], [7], [13], and [8], etc. ).…”

(Generalized) filter properties of the amalgamated algebra

“…Recently, many properties of amalgamations investigated in several papers (e.g. [3], [4], [6], [20], etc.) and the construction has proved its worth providing numerous (counter)examples in commutative ring theory.…”

“…This structure is a well-known structure in commutative algebra and have applications in algebraic geometry. Then, in [17], Nhan extended this notion to generalized regular sequence, which in turn, It has already investigated that when R f J is one of the three first in the above list ( [2], [4], [5], [6]). In this paper, we investigate when it is one of the two last properties.…”