Just give me a hint, since this is assessment! DO NOT TELL ME THE IDEAL
I want to find the left (or right) ideals of the ring of $n\times n$ complex valued matrices.
Now the definition is (for left ideals):
A subset $I$ of $R$ is called a left ideal of $R$ if it is an additive subgroup of $R$ absorbing multiplication on the left:
$$(I,+) \text{ is a subgroup of } (R,+)$$ $$\forall x \in I, \forall r\in R:\quad r\cdot x \in I$$
Now my problem is: We don't start with any element of $I$ from the definition, so we can't iteratively determine the ideal. So my next assumption is guessing at the ideal. Let the ideal be the entire group. Then clearly $r\cdot r \in I$. So then the left ideal is equal to the whole ring. Where is my idea wrong?