course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Groups, Rings and Modules |
IB |
33 |
|
Paper 4, Section II, E |
2014 |
(a) Consider the four following types of rings: Principal Ideal Domains, Integral Domains, Fields, and Unique Factorisation Domains. Arrange them in the form
Prove that these implications hold. [You may assume that irreducibles in a Principal Ideal Domain are prime.] Provide examples, with brief justification, to show that these implications cannot be reversed.
(b) Let