Generalized IP-Injective Rings
Received:July 16, 2003  
Key Word: $S$-$IP$-injective ring   simple-injective ring   $C2$-ring.  
Fund ProjectL:the Specialized Research Fund for the Doctoral Program of Higher Education of China (20020284009, 20030284033), the Postdoctoral Research Fund of China (2005037713), Jiangsu Planned Projects for Postdoctoral Research Fund (0203003403).
Author NameAffiliation
MAO Li-xin Dept. of Basic Courses, Nanjing Institute of Technology, Jiangsu 210013, China 
TONG Wen-ting Dept. of Math., Nanjing University, Jiangsu 210093, China 
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      For a ring $R$, let $ip(R_{R})=\{a\in R$: every right $R$-homomorphism $f$ from any right ideal of $R$ into $R$ with $Imf=aR$ can extend to $R$\}. It is known that $R$ is right $IP$-injective if and only if $R=ip(R_{R})$ and $R$ is right simple-injective if and only if $\{a\in R: aR$ is simple\} $\subseteq ip(R_{R})$. In this note, we introduce the concept of right $S$-$IP$-injective rings, i.e., the ring $R$ with $S\subseteq ip(R_{R})$, where $S$ is a subset of $R$. Some properties of this kind of rings are obtained.
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