… domains, and the (1,1)-domains are Prüfer domains [4]. A commutative … the amalgamated
duplication of a ring A along an A-submodule E of Q(A) (the total ring of fractions of the ring A) …

… when a bi-amalgamation is an AV -ring. Our results capitalize … , we provide necessary and
sufficient conditions for a pair (R, … a semilocal Prüfer domain with exactly two maximal ideals M …

… amalgamated algebras. We will find the necessary and sufficient conditions for the ring A ▷◁f
… If f−1(J) = 0, the following theorem will allow us to determine conditions for A ▷◁f J to be …

… of the amalgamated duplication of a ring along an ideal … on the transfer of various ideal and
ring-theoretic properties from A … and sufficient conditions for which each proper ideal of A ◃▹f …

… of the amalgamated duplication of a ring along an ideal, denoted … Examples of such rings
are the ϕ-Prüfer rings, the PF-rings, … and sufficient condition to have the converse. Recall that a …

… amalgamated algebra (or amalgamation) of R with S along J with respect to f. This construction
generalizes the amalgamated duplication of a ring along an ideal (… we provide conditions …

… , the (0, 1)-domains are the Dedekind domains, and the (1, 1)-domains are the Prüfer domains
[7]. A … , A or f(A) + J is a non-semisimple, and at least one of the following conditions holds: …

… of the amalgamated duplication of a ring along an ideal (… rings and \(\phi \)-PVRs in the
amalgamated algebra along … following two conditions are equivalent for a principal ideal \(…

… n-absorbing ideals and strongly n-absorbing ideals in the amalgamation of A with B along J
… (4) Assume that A is a Prüfer domain and let K := I ⊳⊲f J be an ideal of A ⊳⊲f J, where I is an …

… necessary and sufficient conditions for the graded amalgamation A ⊳⊲ f J to be a graded-coherent
ring. Our … , graded valuation domains and graded-Prüfer rings. For instance see [3]. …