2013-35.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	35	IB 2013 Groups, Rings and Modules	Paper 2, Section II, G	2013

(i) State the structure theorem for finitely generated modules over Euclidean domains.

(ii) Let $\mathbb{C}[X]$ be the polynomial ring over the complex numbers. Let $M$ be a $\mathbb{C}[X]$ module which is 4-dimensional as a $\mathbb{C}$-vector space and such that $(X-2)^{4} \cdot x=0$ for all $x \in M$. Find all possible forms we obtain when we write $M \cong \bigoplus_{i=1}^{m} \mathbb{C}[X] /\left(P_{i}^{n_{i}}\right)$ for irreducible $P_{i} \in \mathbb{C}[X]$ and $n_{i} \geqslant 1$.

(iii) Consider the quotient ring $M=\mathbb{C}[X] /\left(X^{3}+X\right)$ as a $\mathbb{C}[X]$-module. Show that $M$ is isomorphic as a $\mathbb{C}[X]$-module to the direct sum of three copies of $\mathbb{C}$. Give the isomorphism and its inverse explicitly.