Global dimension of bi-amalgamated algebras along pure ideals

Let f : A → B and g : A → C be two ring homomorphisms and let J and J′ be two ideals of B and C, respectively, such that f−1(J) = g−1(J′). The bi-amalgamation of A with (B, C) along (J, J′) with respect to (f, g) is the subring of B × C given byA⋈f,g(J,J′)={(f(a)+j,g(a)+j′)∣a∈A,(j,j′)∈J×J′}A⋈f,g(J,J′)={(f(a)+j,g(a)+j′)∣a∈A,(j,j′)∈J×J′}The aim of this paper is to characterize the global dimension of bi-amalgamated algebras over pure ideals.