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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2020 • Question 10

Question

Math 6310 • Algebra Spring 2020 • Question 10

For each positive integer \(n\) let \(R_n=\mathbb {Z}[2^{1/2},\ldots ,n^{1/n}]\) viewed as a subring of the field of real numbers. Prove that the ring \(R:=\bigcup _{n\ge 1}R_n\) is not Noetherian.

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For each positive integer \(n\) let
\(R_n=\mathbb{Z}[2^{1/2},\ldots,n^{1/n}]\) viewed as a subring of the field of real numbers.
Prove that the ring \(R:=\bigcup_{n\ge 1}R_n\) is not Noetherian.

Topic Tags

Polynomial Rings and Ring Homomorphism Kernels