6310-2023-01

Let \(k=\mathbb {Z}/3\mathbb {Z}\) be the field with three elements. Find all the prime ideals in the ring \(R=k[x,y]/(x^3-x,y^2+x)\).

Let \(k=\mathbb {Z}/2\mathbb {Z}\) and \(R=k[x]\). Let \(M\) denote the cokernel of the map \(R^3\to R^3\) given by the matrix: \[ \begin {bmatrix} 1+x&x^2&1+x\\ 1&x&1\\ 1&x^2&1+x^2 \end {bmatrix} \]

Write \(\operatorname {Hom}(M,R/(x))\) as a direct sum of cyclic modules.

Let \(R=(\mathbb {Z}/3\mathbb {Z})[x,y]\) and let \(I=(x,2y)\). Compute the number of elements in \(\operatorname {Ext}^i(R/I,R)\) for each \(i\ge 0\).

Suppose \(R\) is a commutative ring and \(I\) is an ideal. Let

Prove directly that \(J\) is an ideal of \(R\).

Determine up to similarity, all real \(5\times 5\)-matrices with characteristic polynomial \(x(x^2+1)^2\).