Show that the only bi-lateral ideal of $M_2(\mathbb{C})$ is $\{0\}$. I can see that $\{0\}$ is a bi-lateral (two-sided) ideal but, I cannot show that it is the ONLY one because even if I tried applying left and right actions to an arbitrary matrix, it seems this wouldn't work. Any feedback would be great.
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Consider the action of $GL_2(\mathbb{C})$ on $M_2(\mathbb{C})$ by conjugation. – anomaly Feb 28 '22 at 16:23
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@anomaly If I consider that, it's just something like $AMA^{-1}$ and then we have things like $A(M+N)A^{-1}=AMA^{-1}+ANA^{-1}$ but, it's still hard to see why only $M=0$ fits the bill.. – NoodleNami Feb 28 '22 at 16:29
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Well, there's also $M_2(\mathbb{C})$ itself. – anomaly Feb 28 '22 at 17:17
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@NoodleNami Feedback: had your question not been a duplicate, it would have been closed for quality issues. Please see this advice for improving questions, and also try to exercise using the site's search engine. – rschwieb Feb 28 '22 at 18:26