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Let $R$ be a ring without identity and $n$ be a positive integer. Prove that the ring $R$ is simple iff $M_n(R)$ is simple.

When $R$ is unital ring (i.e. with identity) it is easy because we know the structure of two-sided ideals in $M_n(R)$. So we can consider the case, when $R$ is non-unital.

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  • $\begingroup$ this question differs from math.stackexchange.com/questions/654451/… because the last is for unital rings $\endgroup$ – Mikhail Goltvanitsa Dec 8 '17 at 19:48
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    $\begingroup$ What exactly is your definition of simple rings, please? $\endgroup$ – rschwieb Dec 8 '17 at 20:13
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    $\begingroup$ $ R $ is simple if $ R^2\neq 0$ and there are no non-trivial two-sided ideals in $ R $. $\endgroup$ – Mikhail Goltvanitsa Dec 9 '17 at 4:11