There is a famous topological result:

Let $X$ be a smooth manifold of dimension $n$, $E$ be a vector bundle of rank $k > n$, then $E$ contains a trivial line bundle.

So, I guess that (enlightened by Hartshorne's Exercise 2.8.2):

Let $A$ be a ring (some more conditions are needed, say Noetherian, universally catenary,...) of Krull dimension $n$ and let $P$ be a projective module of rank $k > n$. Then there exists a split injection $A \hookrightarrow P$

I wonder if there's a result similar to this one.

direct summandof $M$. Since a projective module is torsion free, any nonzero element gives rise to an injection $A\hookrightarrow M$. $\endgroup$5more comments