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A ring $R$ is simple when the only ideals of $R$ are $R$ itself and the trivial $\{0\}$.
A matrix ring is any ring of $n\times n$ matrices where multiplication is the usual matrix multiplication, this ring can be over any ring or field.

Is a matrix ring always simple ? I think it should be simple and I am looking for a rigorous proof.

The following answer assume the ring is unital and is also not related immediately to my question
for any ring $A$ the matrix ring $M_n(A)$ is simple if and only if $A$ is simple

where when multiplying by elementary matrices we need to check that these matrices are in our 'smaller' ring and not in the bigger $M_n(A)$ ring
A matrix ring can just be a proper subring of $M_n(A)$ for example. Or is it not possible ?

Then what if also our ring is not unital ?

NotaChoice
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1 Answers1

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I think you are saying a matrix ring to mean "a subring of $M_n(R)$ for some ring $R$ and positive integer $n$".

Clearly not all subrings could be simple. You could take, for example, the subring consisting of just the matrices which are zero off the diagonal, which is never simple if $n>1$.

rschwieb
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  • Thank you, can you come up with a counter example for this statement: "Let A be a unital Banach algebra, let n be a positive integer, and let $\phi : A \rightarrow M_n$ be a homomorphism of complex algebras, $M_n$ denoting the algebra of all $n\times n$ matrices over $\mathbb{C}$. Show that $\phi$ is continuous " ? which is clear if we add the assumption that $\phi$ is surjective, I am looking for a counter example if we omit this assumption. – NotaChoice Feb 14 '22 at 02:21