Basic property of a tensor product I think that this might follow from a basic properties of tensor products, but I am q bit stuck...
Let $A$ be a $k$-algebra. Let $l/k$ be a finite field ext. of $k$. Suppose 
$A \otimes_k l$ is an integral domain. Does it follow that $A \rightarrow A \otimes_k l$
defined by $a$ to $a \otimes 1$ is injective? Thank you!

If $V$ and $W$ are two $k$-vector spaces, and $w\in W$ is nonzero, then the map $V\to V\otimes_k W$ sending $v$ to $v\otimes w$ is always injective, because vector spaces all have a basis, and are therefore flat.
