This note demonstrates the equivalence of S-coherence between rings and finitely presented modules, clarifying a prior question.
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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Xiaolei Zhang (2026) studied this question.
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