| course |
course_year |
question_number |
tags |
title |
year |
Groups, Rings and Modules |
IB |
33 |
IB |
2021 |
Groups, Rings and Modules |
|
Paper 3, Section II, 10G |
2021 |
Let $p$ be a non-zero element of a Principal Ideal Domain $R$. Show that the following are equivalent:
(i) $p$ is prime;
(ii) $p$ is irreducible;
(iii) $(p)$ is a maximal ideal of $R$;
(iv) $R /(p)$ is a field;
(v) $R /(p)$ is an Integral Domain.
Let $R$ be a Principal Ideal Domain, $S$ an Integral Domain and $\phi: R \rightarrow S$ a surjective ring homomorphism. Show that either $\phi$ is an isomorphism or $S$ is a field.
Show that if $R$ is a commutative ring and $R[X]$ is a Principal Ideal Domain, then $R$ is a field.
Let $R$ be an Integral Domain in which every two non-zero elements have a highest common factor. Show that in $R$ every irreducible element is prime.