A commutative unital ring is a field precisely when the zero ideal is maximal

Let R be a commutative ring with 1 \neq 0. Prove that R is a field if and only if 0 is a maximal ideal.


(\Rightarrow) Suppose R is a field. Let I \subseteq R be an ideal which properly contains 0; then there exists an element a \neq 0 in I. Since R is a field, a is a unit in R. By Proposition 9, I = R. Thus R is the only ideal of R which properly contains 0, and hence 0 is a maximal ideal in R.

(\Leftarrow) Suppose 0 is a maximal ideal in R. Now R is commutative by hypothesis. Let a \in R be nonzero. Then since 0 is maximal, we have (a) = R. Since 1 \in R, there exist elements b,c \in R such that ab = ca = 1. By §7.1 #28, a has a two-sided inverse. Thus every nonzero element of R is invertible, and hence R is a field.

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