广义IP-内射环

对环R,令ip(R_R)={a∈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∈R：aR is simple)(?)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(?)ip(R_R)的环。并得到了这种环的一些性质。

Generalized IP-Injective Rings

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.