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I am trying to show that the only ideals in M (2, ## \mathbb R ##) , the ring of 2x2 matrices with Real entries are the trivial ones.

I have a proof, but I am being kind of lazy rigorizing it. We know we cannot have any matrix in GL(n,##\mathbb R ##), because we can then get the identity and we end up with the whole ring. Basically then, we take any

non-invertible matrix m and we show we can find matrices A,A' in M (2, ## \mathbb R ##) so that Am + Am' is invertible. Is there a way of tightening this?

I thought of using the result that maybe either GL(n, ## \mathbb R ## ), or maybe

M (2,##\mathbb R ##) acts transitively on the left on M (2, ## \mathbb R ##) by multiplication. Is this result true?

Thanks.

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# Triviality of Ideals in M(n,R).

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