Let R be a commutative ring with unity 1R such that |R| ≥ 2. (a) Prove that {0} is a prime ideal of R ⇔ R is an integral domain. (b) Prove that {0} is the unique maximal ideal of R ⇔ R is a field. (c) Assume that |R| < ∞. Prove that for an ideal I of R, I is a prime ideal of R ⇔ I is a maximal ideal of R.
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(a) We want to show that {0} is a prime ideal of R if and only if R is an integral domain. Show more…
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