suppose we are given a commutative ring $R$, a principal ideal $I$ and an injective ring map $R \stackrel{f}{\longrightarrow} R^n$, which on quotients $R/I \longrightarrow (R/I)^n$ is the inclusion of a direct summand.
Question: Can one show that $f$ has to be the inclusion of a direct summand?
If this is too general, what are necessary conditions for something like this to be true?
Thanks.