*Note on bi-Amalgamated modules along ideals*

**Adam Anebri **

Laboratory of Modelling and Mathematical Structures Department of Mathematics,

Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Morocco.

Pages 83-92 | Received 20 November 2022, Accepted 20 January 2023, Published 16 June 2023

**Abstract**

Let f: A\rightarrow B and g: A\rightarrow C be two commutative 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 with (B, C) along (J, J') with respect to (f,g), denoted by A\bowtie^{f,g}(J,J'), is the subring of B\times C given by A\bowtie^{f,g}(J,J'):= \{(f(a)+j,g(a)+j') \mid a\in A, (j,j')\in J\times J'\}. In this paper, we study some basic properties of a special kind of A\bowtie^{f,g}(J,J')-modules,called the bi-amalgamation of M with (N,P) along (J,J') with respect (\varphi,\psi), and defined by M \bowtie^{\varphi,\psi}(JN,J'P) := \{(\varphi(m)+n,\psi(m) + p)\ |\ m\in M \text{ and } (n,p) \in JN\times J'P\}. The new results generalize some known results on the bi-amalgamation of rings and the amalgamation of modules along an ideal.

**Keywords**: Bi-amalgamation, amalgamation, Noetherian module, prime module, reduced module, coherent module.

**MSC numbers**: 13E05, 13D05, 13D02, 13C60.

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