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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