What question did this study set out to answer?

This work aims to explore the relationship between S-coherent rings and finitely presented S-modules.

February 28, 2026

A note on S -coherent rings

Puntos clave

This work aims to explore the relationship between S-coherent rings and finitely presented S-modules.
Analyzed the definition of S-coherence in ring theory.
Provided examples to illustrate the properties of S-coherent rings and modules.
Investigated the implications of S-coherence for finitely presented modules.
Established that a ring is S-coherent if every finitely presented S-module is S-coherent.
Demonstrated that S-S-coherent rings are S-coherent, but not vice versa through counterexample.

Resumen

In this note, we show that a ring Formula: see text is Formula: see text-coherent if and only if every finitely presented Formula: see text-module is Formula: see text-coherent, providing a positive answer to a question proposed in D. Bennis, M. El Hajoui, On [Formula: see text-coherence, J. Korean Math. Soc. 55 (2018), no.6, 1499-1512]. Besides, we show that Formula: see text-Formula: see text-coherent rings are Formula: see text-coherent, and give an example to show the converse is not true in general.

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