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Pick out the true statements:

a. Let $R$ be a commutative ring with identity. Let $M$ be an ideal such that every element of $R$ not in $M$ is a unit. Then $R/M$ is a field.

b. Let $R$ be as above and let M be an ideal such that $R/M$ is an integral domain. Then $M$ is a prime ideal.

c. Let $R = C[0, 1]$ be the ring of real-valued continuous functions on $[0, 1]$ with respect to pointwise addition and pointwise multiplication. Let $M = \{ f ∈ R \mid f(0) = f(1) = 0 \}$. Then $M$ is a maximal ideal.

Certainly (b) is true but no idea about (a) and (c).

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1 Answer 1

up vote 2 down vote accepted

Hints:

(a) For this one, use the fact that $R/M$ is a field $\iff$ $M$ is a maximal ideal. What happens if an ideal $I$ in $R$ contains a unit?


(c) For this part, remember that any maximal ideal is also a prime ideal.

Your ideal $M = \{f ∈ R \mid f(0) = f(1) = 0 \}$ is just the set of continuous function $f: [0, 1] \to \mathbb{R}$ such that $f(0) = f(1) = 0$.

Now, try to think of two continuous functions $f, g \in R$ such that their product satisfies $f(0)g(0) = f(1)g(1) = 0$, that is, $f\cdot g \in M$. Does this mean that $f \in M$ or $g \in M$?

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