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Let $R$ be a ring (possibly non-commutative).

Definition $R$ is called a local ring if it has a unique left(and equivalently right) maximal ideal.

I am looking for an example of a ring (obviously non-commutative) which has a unique two-sided maximal ideal but is not local.

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The ring of $2×2$ matrices over reals is simple so it has one maximal two sided ideal, 0, but it has 2 maximal left (right) ideals.

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    $\begingroup$ +1 Note: “has two maximal right ideals” is true, but is quite an understatement: this ring has infinitely many distinct maximal right ideals. $\endgroup$ – rschwieb Jul 18 '20 at 21:58
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    $\begingroup$ The OP asked for two. There are uncountably many maximal right (left) ideals: cambridge.org/core/journals/canadian-journal-of-mathematics/… $\endgroup$ – Mark Sapir Jul 18 '20 at 22:04
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    $\begingroup$ I know they asked for “more than one”, and the note is certainly not to be interpreted as pointing out a fault, it’s just that some people are under the misapprehension that this ring has exactly two maximal right ideals. The remark hedges against causing that confusion. $\endgroup$ – rschwieb Jul 18 '20 at 22:55
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    $\begingroup$ The text I linked to contains a description of all maximal left ideals of the ring of matrices. $\endgroup$ – Mark Sapir Jul 18 '20 at 22:59
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Literally every simple ring that isn’t a division ring, so for example every matrix ring over a division ring ($n>1$ of course), and the first Weyl algebra.

Beyond those also, every ring of linear transformations of an infinite dimensional vector space has a unique maximal two sided ideal, but isn’t local (and also isn’t simple). In fact their two sided ideals are linearly ordered.

And if you have any maximal ideal $M$ of a ring $R$ that you know isn’t maximal as a right ideal, and it isn’t nilpotent, then you can force it to be a unique maximal ideal in the quotient ring $R/M^n$, which won’t be local because of right ideal correspondence.

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  • $\begingroup$ "Literally every simple ring that isn’t a division ring" means "all of them" or "literally all of them"? $\endgroup$ – Mark Sapir Jul 19 '20 at 14:45
  • $\begingroup$ @JCAA Again, thanks for catching the omission, but I'm afraid I don't have time to play your games. If there's another correction you're suggesting you'll have to say it straight. $\endgroup$ – rschwieb Jul 19 '20 at 14:53
  • $\begingroup$ You did not answer my question. Is that straight enough? $\endgroup$ – Mark Sapir Jul 19 '20 at 14:55
  • $\begingroup$ Thank you for taking the time to edit the insult out of your last comment. For the sake of getting you off the topic of grammar and to your point, let's say that I think that in natural language "Literally every A that isn't a B" is the same sentence if "literally" is omitted. Now your turn again: where's this going? $\endgroup$ – rschwieb Jul 20 '20 at 18:28
  • $\begingroup$ @JCAA Just noticed I forgot to tag you in the last comment. You were saying? $\endgroup$ – rschwieb Jul 21 '20 at 17:32

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