On S-coherent property in amalgamated algebra along an ideal
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The authors in \cite{BE} have defined a (commutative unital) ring $R$ to be $S$-coherent if each finitely generated ideal of $R$ is an $S$-finitely presented $R$-module. Any coherent ring and any $S$-Noetherian ring is a $S$-coherent ring. In this paper, we investigate the transfer of the property of $S$-coherence to the amalgamation $A\bowtie^{f}J$. We provide necessary and sufficient conditions for $A\bowtie^{f}J$ to be an $S$-coherent ring.
Abstract
The authors in \cite{BE} have defined a (commutative unital) ring $R$ to be $S$-coherent if each finitely generated ideal of $R$ is an $S$-finitely presented $R$-module. Any coherent ring and any $S$-Noetherian ring is a $S$-coherent ring. In this paper, we investigate the transfer of the property of $S$-coherence to the amalgamation $A\bowtie^{f}J$. We provide necessary and sufficient conditions for $A\bowtie^{f}J$ to be an $S$-coherent ring.