Let $K$ be a field and $R=K^{2 \times 2}$.
Let $U$ be a subspace of $\mathbb{Q}^2$ and $L=\{A \in R: coim(A) \subset U\}$. Then $L$ is a left ideal of $R$.
For $L$ to be a left ideal the difference of two elements of $L$ has to be in $L$ as well. This should be clear because $U$ is a subspace which has exactly this property (?)
The second property is that for every $r \in R$, $r \cdot l \in L$. This is a similar property of a subspace however I struggle to find the connection to the "co-image" of $A$ to correctly finish the proof. I couldn't find an understandable definition of the "co-image".
I also need to prove the reverse direction: If $L$ is a left ideal of $R$ then there exists a subspace $U$ of $\mathbb{Q}^2$ so that $L=\{A \in R: coim(A) \subset U \}$.
Thanks for any help!