Bulletin of the Iranian Mathematical Society (2021) 47:1599–1609 https://doi.org/10.1007/s41980-020-00461-y ORIGINAL PAPER Amalgamated Algebras Issued from -Chained Rings and -Pseudo-Valuation Rings Abdelhaq El Khalfi 1 · Hwankoo Kim 2 · Najib Mahdou 1 Received: 25 April 2020 / Revised: 17 August 2020 / Accepted: 31 August 2020 / Published online: 24 September 2020 © Iranian Mathematical Society 2020 Abstract Let f : A −→ B be a ring homomorphism and let J be an ideal of B . In this article, we study the possible transfer of the properties of being a φ-ring, a φ-chained ring, and a φ-pseudo-valuation ring to the amalgamated algebra A ⊲⊳ f J . Our aim is to provide examples of new classes of commutative rings satisfying the aforementioned properties. Keywords Amalgamated algebra · Chained ring · Pseudo-valuation ring · Trivial ring extension · Valuation ring Mathematics Subject Classification 13B99 · 13A15 · 13G05 · 13B21 1 Introduction Throughout this paper, all rings are commutative with identity and all modules are unital. If R is a ring, we denote by Nil( R) and J( R) the ideal of all nilpotent elements of R and the Jacobson radical of R, respectively. Hedstrom and Houston [19] introduced the class of pseudo-valuation domains, which is closely related to the class of valuation domains. A domain R with quotient field K is called a pseudo-valuation domain (in Communicated by Siamak Yassemi. B Hwankoo Kim hkkim@hoseo.edu Abdelhaq El Khalfi abdelhaq.elkhalfi@usmba.ac.ma Najib Mahdou mahdou@hotmail.com 1 Modelling and Mathematical Structures Laboratory, Department of Mathematics, Faculty of Science and Technology of Fez, University S.M. Ben Abdellah Fez, Box 2202, Fez, Morocco 2 Division of Computer and Information Engineering, Hoseo University, Asan, South Korea 123