Communications of the
Korean Mathematical Society CKMS

Let $R$ be a commutative ring with identity. If the nilpotent radical $Nil(R)$ of $R$ is a divided prime ideal, then $R$ is called a $\phi$-ring. Let $R$ be a $\phi$-ring and $S$ be a multiplicative subset of $R$. In this paper, we introduce and study the class of nonnil-$S$-coherent rings, i.e., the rings in which all finitely generated nonnil ideals are $S$-finitely presented. Also, we define the concept of $\phi$-$S$-coherent rings. Among other results, we investigate the $S$-version of Chase's result and Chase Theorem characterization of nonnil-coherent rings. We next study the possible transfer of the nonnil-$S$-coherent ring property in the amalgamated algebra along an ideal and the trivial ring extension.

Keywords: Nonnil-$S$-coherent ring, $\phi$-$S$-coherent ring, $S$-coherent ring, nonnil-coherent ring, $\phi$-$S$-flat module

MSC numbers: Primary 13Axx, 13Bxx, 13Cxx