广义IP-内射环

Generalized IP-Injective Rings

中文关键词:  $S$-$IP$-内射环  单-内射环  $C2$-环.

英文关键词:$S$-$IP$-injective ring  simple-injective ring  $C2$-ring.

中文摘要:

      对环$R$,令$ip(R_{R})=\{a\in R$:任意一个从$R$的右理想到$R$且象为$aR$ 的模同态能开拓到$R$\}. 众所周知,$R$为右$IP$-内射环当且仅当$R=ip(R_{R})$, $R$为右单-内射环当 且仅当$\{a\in R: aR$ is simple\} $\subseteq ip(R_{R})$. 对环$R$的一个子集$S$, 我们引进了 $S$-$IP$-内射环的概念,即满足$S\subseteq ip(R_{R})$的环. 并得到了这种环的一些性质.

英文摘要:

      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.