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There is a famous topological result:

Let $X$ be a smooth manifold of dimension $n$, $E$ be a vector bundle of dimension $k > n$, 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 an split injection $A \hookrightarrow P$

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

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.

There is a famous topological result:

Let $X$ be a smooth manifold of dimension $n$, $E$ be a vector bundle of dimension $k > n$, 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 an injection $A \hookrightarrow P$

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

There is a famous topological result:

Let $X$ be a smooth manifold of dimension $n$, $E$ be a vector bundle of dimension $k > n$, 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 an split injection $A \hookrightarrow P$

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

There is a famous topological result:

Let $X$ be a smooth manifold of dimension $n$, $E$ be a vector bundle of dimension $k > n$, 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 needed, say Noetherian, universally catenary,...) of Krull dimension $n$, $P$ be a projective module of rank $k > n$. Then there exists an injection $A \hookrightarrow P$

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

There is a famous topological result:

Let $X$ be a smooth manifold of dimension $n$, $E$ be a vector bundle of dimension $k > n$, 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 an injection $A \hookrightarrow P$

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