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We know that a simple ring has characteristic either $ 0 $ or a prime $ p $. I am thinking about giving a concrete example of a simple ring WITH zero divisors.

I have found a reference talking about simple rings without zero-divisors:

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/438A61A4C89B22ADF1E46E759C3A7E3D/S002557930000187Xa.pdf/div-class-title-simple-rings-without-zero-divisors-and-lie-division-rings-div.pdf

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A matrix ring $M_n(R)$ when $R$ is a simple ring, is also simple. For instance one can take $R$ to be one's favourite field. Matrix rings generally have zero divisors.

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