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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2001 • Question 9

Question

Math 6310 • Algebra Fall 2001 • Question 9

Show that \(\mathbb {Q}\otimes _{\mathbb {Z}}\mathbb {Q}\cong \mathbb {Q}\), and that \(\mathbb {Q}\otimes \mathbb {Z}/N\mathbb {Z}=0\) for every positive integer \(N\), while \(\mathbb {Q}\otimes \left (\prod \mathbb {Z}/N\mathbb {Z}\right )\ne 0\), where the product is taken over all positive integers \(N\).

Raw LaTeX

Show that \(\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{Q}\cong\mathbb{Q}\), and that
\(\mathbb{Q}\otimes \mathbb{Z}/N\mathbb{Z}=0\) for every positive integer \(N\), while
\(\mathbb{Q}\otimes\left(\prod \mathbb{Z}/N\mathbb{Z}\right)\ne 0\), where the product is taken
over all positive integers \(N\).

Topic Tags

Tensor Products of Modules