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Show the ring $A=\mathrm{Mat}(F)$ has exactly two 2-sided ideals, even though it is not a division ring.

$F$ is any field, and $\mathrm{Mat}(F)$ is all $n\times n$ matrices with elements of $F$ as entries. I don't understand how this is true. If you have a $2 \times 2$ matrix, and you have only 1 entry being nonzero (say $e_{11}$, the top left entry), wouldn't the ideal generated by this element be only this matrix multiplied by some coefficient? Also, why is it not a division ring even though it has two 2-sided ideals (i know its not a division ring since not all matrices are invertible, but what condition is not met?)

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  • $\begingroup$ the ring $\,A\,$ is simple in this case, and the only two-sided ideals are the trivial ones: the zero ideal and the whole ring. $\endgroup$ – DonAntonio Apr 20 '13 at 0:55
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You can "move around" the single non-zero entry via row and column operation matrices; since ideals are closed under scaling and addition, you can generate everything.

Also, I'm not sure what you're confused by in your second question - there are non-zero elements of $\mathrm{Mat}(F)$ that are not invertible; a division ring is a ring in which every non-zero element is invertible; therefore $\mathrm{Mat}(F)$ is not a division ring.

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  • $\begingroup$ Well, I suppose once the OP is satisfied that there are only two 2-sided ideals, this will provide a counterexample to that belief. $\endgroup$ – Zev Chonoles Apr 20 '13 at 0:40
  • $\begingroup$ its because one of my hw problems stated that a the conditions that a ring having 2 right ideals, a ring having 2 left ideals, and a division ring are equivalent. Is that false? $\endgroup$ – 010110111 Apr 20 '13 at 0:40
  • $\begingroup$ @stephan: No, that's not false. But why do you think either the condition "has exactly two left ideals" or the condition "has exactly two right ideals" is the same as "has exactly two 2-sided ideals"? The statement "has exactly two 2-sided ideals" does nothing to rule out the possibility that there are more things that are left ideals but not right ideals, or right ideals but not left ideals. $\endgroup$ – Zev Chonoles Apr 20 '13 at 0:43
  • $\begingroup$ Ok got it! thanks for the explanation $\endgroup$ – 010110111 Apr 20 '13 at 0:45

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