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Let $\mathrm{R}$ be a commutative ring with unity. Which of the following is true?

1. If $\mathrm{R}$ has finitely many prime ideals, then $\mathrm{R}$ is a field

2. If $R$ has finitely many ideals, then $R$ is finite

3. If $R$ is a P.I.D., then every subring of $R$ with unity is a P.I.D.

4. If $\mathrm{R}$ is an integral domain which has finitely many ideals, then $\mathrm{R}$ is a field
in Abstract Algebra by Expert (2.4k points)

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