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In a unital ring, if the quotient corresponding to an ideal is a field, then the ideal is maximal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.5

Solution: Suppose we have a two-sided ideal $I$ with $M \subseteq I \subseteq R$. By the Lattice Isomorphism Theorem for rings, $I/M$ is a two-sided ideal of the field $R/M$. In particular, $R/M$ is commutative, and so by Proposition 9, the only ideals of $R/M$ are $M/M$ and $R/M$. Again using the Lattice Isomorphism Theorem, we have $I = M$ or $I = R$. So $M$ is maximal in $R$.


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