# I.1.9

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This problem has Harthshorne Height 1.

This problem has height 1.

### HAPPY

• The problem translates to showing if $P \supset a$ is prime, then $ht(P)\le r$.
• Recall $\sqrt{a} = \cap_i P_i$ for certain minimal primes, use the assumptions to show that each $P_i$ is generated by at most r elements.
• Use commutative algebra below to finish.

### Commutative Algebra

• If you have a Noetherian local ring $A$ with maximal ideal $m$ then $ht\ m = \dim A \le$ the number of generators of $m$.