The critical step in the proof of Baer's Criterion involves defining a map $g(r)=f(rx)$ on the constructed ideal. I think this bears a similarity with extending a map from $\mathbb R^n$ to $\mathbb R^{n+1}$, for example: if $\vec x$ is any vector not in $\mathbb R^n$, we can first find a value for $\vec x$, and then extend it to the whole line spanned by $\vec x$.
Is this intuition justified? Are there similar circumstances in geometry/algebraic geometry? If this is indeed "geometric," why would we think of using this strategy in the proof? (I find it hard to relate injective modules to any kind of geometry.)
