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 ⨝ f J was introduced by M. D’Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of ( R ⨝ f J )-module called the amalgamation of M and N along J with respect to φ, and denoted by M ⨝ φ JN . We study some homological properties of the ( R ⨝ f J )-module M ⨝ φ JN . Among other results, we investigate projectivity, flatness, injectivity, Cohen-Macaulayness, and prime property of the ( R ⨝ f J )-module M ⨝ φ JN in connection to their corresponding properties of the R -modules M and JN .