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I saw "Any simple ring is a prime ring" as an example in Prime ring@wiki.

Can anyone show me how to proof it?

Also, on the other side, is any prime ring a simple ring?

Thanks.

Richard
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  • For the second question, not all prime rings are simple. Consider any integral domain which is not a field, e.g. the integers. – TomGrubb Apr 11 '15 at 03:47
  • What if it is finite? – Richard Apr 11 '15 at 03:51
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    All finite integral domains are fields. See here for more: http://math.stackexchange.com/questions/62548/why-is-a-finite-integral-domain-always-field – TomGrubb Apr 11 '15 at 05:48

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A unital ring $R$ is called simple iff $R$ is not zero ring and the only ideals of $R$ are $R$ and $0$; prime iff $R$ is not zero ring and for any nonzero ideals $I,J$ of $R$, $IJ\neq 0$. Since the only nonzero ideal of a simpile ring $R$ is $R$ itself, and $R^2=R\neq 0$, it's clear that simple rings are prime rings.

If $R$ is commutative, simple rings = fields, prime rings= integral domains. Not every integral domain is a field, the simplest example is $\mathbb Z$ I think.

Censi LI
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