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I'm reading through Lang's Algebra. Lang defines a simple ring as a semisimple ring that has only one isomorphism class of simple left ideals.

On the other side, Wikipedia says that a simple ring is a non-zero ring that has no two-sided ideals except zero ideal and itself. I'm wondering, are these two definitions equivalent?

Lang shows that his definition is stronger than the one on Wikipedia, but starts Rieffel's theorem as:

Let $R$ be a ring without two-sided ideals except $0$ and $R$...

which raises suspicion about the equivalence.

Notes:

1) For Lang, ring has to have the multiplicative identity.

2) Lang defines semisimple rings as rings who are semisimple as (left) modules over itself, but later shows that left-right distinction doesn't matter.

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The one in Wikipedia is the standard definition of a simple ring, but from what you describe, Lang's definition amounts to what is more commonly called a simple Artinian ring.

For a simple ring (by which I mean the definition appearing in Wikipedia) the conditions of being right Artinian or left Artinian are equivalent. As a special case of the Artin-Wedderburn theorem, such a ring is just $M_n(D)$ for some division ring $D$.

Perhaps it was Lang's intention to define a simple ring such that all of his simple rings are semisimple too. This is a point of terminological confusion sometimes, because the more common definition allows simple rings which aren't Artinian, hence aren't semisimple.

I don't know if anyone has tried it, but maybe it would make sense to call finite products of simple rings "semisimple" and to give "semisimple Artinian rings" another name, say "Wedderburn ring" or something like that. Isaacs called them Wedderburn rings in his Algebra book, actually, but I don't think it caught anywhere else.

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  • $\begingroup$ Would Weyl's algebra be an example of simple ring (in Wiki definition) that isn't semisimple? Are there some other standard examples? $\endgroup$
    – ante.ceperic
    Jul 21 '16 at 15:38
  • $\begingroup$ @ante.ceperic Yes, the Weyl algebra is a well-known example of a simple domain (which most people find to be a bit weird.) $\endgroup$
    – rschwieb
    Jul 21 '16 at 15:39
  • $\begingroup$ @ante.ceperic You can find examples at the DaRT. Actually, I thought of a third one while looking at these. $\endgroup$
    – rschwieb
    Jul 21 '16 at 16:16
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To the condition from Wikipedia, you have to add the condition $R$ is left artinian.

By Wedderburn's theorem, a simple ring is a ring of matrices over a division ring

See Bourbaki, Algebra, ch. 8, Semi-simple modules and rings, §7, no 1, Prop.1.

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