Communications of the
Korean Mathematical Society CKMS

The purpose of this paper is to introduce some new class of rings that are closely related to the classes of sharp domains, pseudo-Dededkind domains, $TV$ domains and finite character domains. A ring $R$ is called a $\phi$-sharp ring if whenever for nonnil ideals $I,A,B$ of $R$ with $I\supseteq AB$, then $I=A'B'$ for nonnil ideals $A',B'$ of $R$ where $A'\supseteq A$ and $B'\supseteq B$. We proof that a $\phi$-Dedekind ring is a $\phi$-sharp ring and we get some properties that by them a $\phi$-sharp ring is a $\phi$-Dedekind ring.

Keywords: $\phi$-sharp ring, $\phi$-pseudo-Dedekind ring, $\phi$-$TV$ ring, $\phi$-finite character ring

MSC numbers: Primary 16N99, 16S99; Secondary 06C05, 16N20