6310-2024-08

Identify all the ideals in the ring \(R=\mathbb {Z}[x]/(6,x^3)\).

Let \(V:=\mathbb {Z}^3\) and \(L\) the submodule of \(V\) spanned by the columns of the matrix \[ A:=\begin {bmatrix} 4&2&2\\ 2&2&6\\ 2&4&4 \end {bmatrix} \]

Find invertible matrices \(Q\) and \(P\) such that the matrix \(QAP\) is diagonal.

Determine all the \(\mathbb {Z}\)-submodules of the \(\mathbb {Z}\)-module \(\mathbb {Z}/(3)\oplus \mathbb {Z}/(3)\oplus \mathbb {Z}/(9)\) consisting of 3 elements.

Determine all the possible Jordan canonical forms for a linear transformation with characteristic polynomial

Prove the following assertions for the ring \(S:=\mathbb {R}[x]/(x^2+1)^2\).

1. There exist exactly two ring homomorphisms \(\pi :S\to \mathbb {C}\) such that \(\pi |_{\mathbb {R}}=\operatorname {id}_{\mathbb {R}}\), the identity map on \(\mathbb {R}\).
2. Choose one of the homomorphisms in (a), and call it \(\pi \). Prove that there exists a unique homomorphism of rings \(\sigma :\mathbb {C}\to S\) such that \(\sigma |_{\mathbb {R}}=\operatorname {id}_{\mathbb {R}}\) and \(\pi \sigma =\operatorname {id}_{\mathbb {C}}\).
3. Prove that \(\sigma \) extends to an isomorphism of rings \(\mathbb {C}[y]/(y^2)\to S\).