2017-29.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	29	IB 2017 Groups, Rings and Modules	Paper 3, Section I, E	2017

Let $R$ be a commutative ring and let $M$ be an $R$-module. Show that $M$ is a finitely generated $R$-module if and only if there exists a surjective $R$-module homomorphism $R^{n} \rightarrow M$ for some $n$.

Find an example of a $\mathbb{Z}$-module $M$ such that there is no surjective $\mathbb{Z}$-module homomorphism $\mathbb{Z} \rightarrow M$ but there is a surjective $\mathbb{Z}$-module homomorphism $\mathbb{Z}^{2} \rightarrow M$ which is not an isomorphism. Justify your answer.