Some homological properties of amalgamated modules along an ideal

Let R and S be commutative rings with identity, J be an ideal of S, f: R → S be a ring homomorphism, M be an R-module, N be an S-module, and let φ: M → N be an R-homomorphism. The amalgamation of R with S along J with respect to f denoted by R ⨝fJ was introduced by M. D’Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of (R ⨝fJ)-module called the amalgamation of M and N along J with respect to φ, and denoted by M ⨝φJN. We study some homological properties of the (R ⨝fJ)-module M ⨝φJN. Among other results, we investigate projectivity, flatness, injectivity, Cohen-Macaulayness, and prime property of the (R ⨝fJ)-module M ⨝φJN in connection to their corresponding properties of the R-modules M and JN.