6310-2022-01

Consider the ideal \(I:=(2x-9,3x-7)\) in the ring \(\mathbb {Z}[x]\). Find the smallest positive integer \(n\) such that

belongs to the ideal \(I\).

Let \(R\) be a commutative ring with identity, and let \(0\to L\to M\to N\to 0\) be an exact sequence of \(R\)- modules. Prove or disprove:

1. If \(M\) and \(N\) are finitely generated \(R\)-modules, then \(L\) is finitely generated.
2. If \(M\) is finitely generated, and \(N\) is a free \(R\)-module, then \(L\) is finitely generated.

Let \(R:=\mathbb {Q}[x]/(x^3-1)\). Give an example of a finitely generated projective \(R\)-module that is not free.

Let \(R\) be a commutative ring with identity such that \(IJ = I \cap J\) for all ideals \(I\) and \(J\). Prove that each prime ideal of \(R\) is maximal.

Let \(M\) be a \(5\times 5\) matrix over the complex numbers \(\mathbb {C}\), such that the eigenvectors of \(M\), along with the zero vector, form a two-dimensional vector subspace of \(\mathbb {C}^{5}\). Determine the possible Jordan forms of \(M\).