On $\phi$-sharp rings
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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.
Abstract
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.
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