What Is the Correct Classification of Rings in Integral Domain Classes?

[FanofAFan]

Homework Statement

Put the following rings in the smallest class: a. Z b.$$Z$$₅ c. Q d. Z₅[x] e. Z f. Z[$$\sqrt{}-5$$] g. Z[x] h. Q[x]

Homework Equations

four class are euclidean $$\subset$$ principal ideal $$\subset$$ unique factorization $$\subset$$ integral

The Attempt at a Solution

What I got for the following was a. euclidean b. euclidean c integral d integral e euclidean f. unique factor g. unique factor h. integral... however I am not sure on b, c, and g please help

 

[mighty2000]

Your attempt at a solution is mostly correct. Here are some additional explanations for the remaining items:

b. Z5 is a field, so it satisfies all four properties and can be placed in the smallest class.
c. Q is also a field, so it satisfies all four properties and can be placed in the smallest class.
g. Z[x] is a unique factorization domain, but not a principal ideal domain. Therefore, it should be placed in the second smallest class (principal ideal).