By Irving Kaplansky

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Instead of quoting Theorem 67, we could use the globalized version: Ex. 17 in $1-6. I 72 CH. 2/NOETHERIAN RINGS SEC. 2-3/INTEGRAL ELEMENTS, Ill 3. An alternative proof of Theorem 98 would begin by quoting Theorem 101, It would then remain to prove T Noetherian, presumably exactly as was done in the proof of Theorem 98. EXERCISES What examples are there of non-Dedekind Priifer domains? The main ones are as follows. (1) Valuation domains. For instance, it is possible to construct a valuation domain with any preassigned value group.

Prove that any variety is expressible as a finite union of irreducible varieties. (Hint: one can derive this directly from the ascending chain condition in K[xl, . ) 20. Let R be an integral domain with quotient field K. Suppose that every ring between R and K is Noetherian. Prove that the dimension of R is at most 1. (Hint: if not, take x # 0, x e Q C P, Q and P distinct primes. Then take y E P, where y is not a member of any prime minimal over x. Let T be the ring generated over R by all xy-*, I the ideal in T they generate.

It shows T/aT to be a submodule of a cyclic R-module, hence a finitely generated R-module. EXERClSES 1. Let A ; be a non-zero R-module for each i ranging over an index set. Let A be the direct sum of all A;. Prove: <(A) = U<(Ai). 2. Given a set-theoretic union UPi of prime ideals in a commutative ring R, prove that UP; = <(A) for a suitable R-module A . 3. Let B be a non-zero R-module, and A a submodule different from 0 or B. Prove: <(A) c c <(’(A) u Z(’(BI-4) 4. Let A be a non-zero module, x an element with x p <(A), x A # A .