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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?

user26857
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hat
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  • Hint: The rows of $rx$ are linear combinations of rows of $x$. You get all such combinations with an appropriate choice of $r$. – Jyrki Lahtonen Apr 29 '15 at 06:24
  • You just assumed that the ideal is the entire ring and verified that,The entire ring is always a left ideal, but there are others... and that's where the interesting game takes place. With right ideals think in terms of column spaces instead. – Jyrki Lahtonen Apr 29 '15 at 06:27
  • Ahhh for some reason I believed there was only 1 left ideal, or 1 right ideal. This makes much more sense – hat Apr 29 '15 at 06:29

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