6310-2021-08

Consider the ring homomorphism given by

Determine a minimal generating set for the kernel.

Let \(R:=\mathbb {Q}[x]\), and let \(M\) be the cokernel of the map \[ R^3\xrightarrow {\begin {pmatrix}x-2&4&x\\0&4&x\\0&-x&-1\end {pmatrix}}R^3. \]

Write \(M\) as a direct sum of cyclic \(R\)-modules.

Prove that \(\mathbb {C}[x]\) is integrally closed in \(\mathbb {C}(x)\), the field of rational functions.

Let \(I\) be the ideal \((2,1+\sqrt {-5})\) in the ring \(R:=\mathbb {Z}[\sqrt {-5}]\), and let \(\Lambda ^2(I)\) denote the second exterior power of \(I\) in the category of \(R\)-modules. Prove or disprove: \(\Lambda ^2(I)\) is zero.

Let \(R\) be a commutative ring. Let \(P\) and \(Q\) be two projective \(R\)-modules. Prove that \(P\otimes _R Q\) is projective.