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What are the two sided maximal ideals of $M_3 (\mathbb {Z})$?

Is the quotient by a maximal ideal a field in case of noncommutative rings?

learning_math
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  • No, it is not. For instance, ${0}$ is the maximal two-sided ideal of $M_n(\Bbb R)$. –  Aug 15 '17 at 01:12
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    The two-sided ideals of $M_n(R)$ (where $R$ is a ring) are all of the form $M_n(I)$ where $I$ is an ideal of $R$. – anon Aug 15 '17 at 01:14
  • Can you point me to a detailed solution or give me some hints? @anon – learning_math Aug 15 '17 at 03:20
  • Google "two-sided ideals of matrix rings." – anon Aug 15 '17 at 03:38
  • Your second question is also a duplicate, although a good dupe is hard to find. You could check out https://math.stackexchange.com/q/1927580/29335 to see why the answer is "no." Such a quotient will always be a simple ring, but not always a field. – rschwieb Aug 15 '17 at 12:26

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