This is a true in a commutative ring with $1$, but does it also hold in a noncommutative ring with $1$? The proof in my book is just an application of Zorn's lemma, but the commutativity of the ring is not used anywhere.
Every proper ideal contained in a maximal ideal?
5
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abstract-algebra
ideals
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9Well, if commutativity is not used in the proof... – 2012-11-29
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9You should be careful when talking about ideals in noncommutative rings: every proper left resp. right resp. two-sided ideal is contained in a maximal left resp. right resp. two-sided ideal. – 2012-11-29