I know that if there exists no ideal different from trivial ideals, the commutative ring with unity is a field. But I cannot figure out any example for a ring has trivial ideals only and not a field.
Thanks in advance
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2Well, if it's a commutative ring with unity, you get a field. So looking at non-commutative rings, or rings without unity might be a good place to start. – Arthur May 23 '19 at 19:09
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2HINT: The noncommutative rings that most people are most familiar with are the rings of matrices over a field. – Arturo Magidin May 23 '19 at 19:12
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Aren't quaternions a decent example? – Randall May 23 '19 at 19:12
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@ArturoMagidin Does $Mat_2(\mathbb R)$ work? – user519955 May 23 '19 at 19:13
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2You tell me; or you figure it out. – Arturo Magidin May 23 '19 at 19:14
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See https://math.stackexchange.com/questions/739205/show-that-m-2-mathbbr-has-no-non-trivial-two-sided-ideals – lhf May 23 '19 at 19:17