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Let ∆ be a simplicial complex with vertex set V = {1, 2, . . , n} and I∆ be the ideal of k[X1 , . . , Xn ] as defined in Q. 14 of Problem Set 2. Consider the residue class ring R∆ := k[X1 , . . , Xn ]/I∆ . Show that dim R∆ = d + 1, where d is the (topological) dimension of ∆. ] 20. Assume that A is a noetherian ring. If a ∈ A is a nonzerodivisor and p is a minimal prime of (a), then prove that ht p = 1. 21. Give an example of a minimal prime p of a principal ideal of a noetherian ring such that ht p = 0.

S. Cohen [Duke Math. J. 17 (1950), pp. 27–42]. In fact, Cohen defines a Dedekind domain to be an integral domain in which every nonzero proper ideals factors as a product of prime ideals, and he notes that the uniqueness of factorization is automatic, thanks to Matsusita [Japan J. Math. 19 (1944), pp. 97–110]. 3 Abstrakter Aufbau der Idealtheorie in algebraischen Zahlund Funktionenk¨orpern, Math. Ann. 96 (1927), pp. 26–61. The Aufbau paper followed another famous paper Idealtheorie in Ringbereichen [Math.

We end this section with a result which gives a sufficient condition for a Dedekind domain to be a PID. 13. A local Dedekind domain is a PID. More generally, a Dedekind domain that has only finitely many maximal ideals is a PID. 50 Proof. Let A be a Dedekind domain with only finitely many maximal ideals, say, P1 , . . , Pr . Note that the ideals P1 , . . , Pr , and more generally, their powers P1m1 , . . , Prmr are pairwise comaximal. Fix any i ∈ {1, . . , r}. Note that Pi = Pi2 (because otherwise Pi = A).