| course |
course_year |
question_number |
tags |
title |
year |
Groups, Rings and Modules |
IB |
33 |
IB |
2010 |
Groups, Rings and Modules |
|
Paper 2, Section II, H |
2010 |
For ideals $I, J$ of a ring $R$, their product $I J$ is defined as the ideal of $R$ generated by the elements of the form $x y$ where $x \in I$ and $y \in J$.
(1) Prove that, if a prime ideal $P$ of $R$ contains $I J$, then $P$ contains either $I$ or $J$.
(2) Give an example of $R, I$ and $J$ such that the two ideals $I J$ and $I \cap J$ are different from each other.
(3) Prove that there is a natural bijection between the prime ideals of $R / I J$ and the prime ideals of $R /(I \cap J)$.