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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 MIR. 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.


Linearity

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