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I'm trying to find all left, right, and two sided ideals in the ring of $n \times n$ complex matrices.

My guess is that the ideals are simply $\{0\}$ and $M_n(\mathbb{C})$ itself, but I'm struggling to prove it. So far, I have concluded that a non-trivial ideal (one that is not the ring itself or the $0$ ring), can't contain any invertible matrices because if it did it would contain the entire ring.

However, I am wondering if it is possible for there to be a non-trivial ideal that consists entirely of non-invertible matrices? How would I show it? Thanks

user1590812
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  • Hint: consider ideals generated by elementary matrices. (Also note that the observation about invertible elements applies to any ring, not particular to matrices.) – Ben May 12 '25 at 06:28
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    You should search the question before asking it. This question has been asked many times in this site. – Yunfeng Gong May 12 '25 at 06:31

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