Generalized IP-Injective Rings

Abstract:

      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.