What kind of subrings inherit their parent ring's properties?

Question

I was studying the proof my teacher from my abstract algebra course gave me for the Gauss Lemma, and I came up with a question he could not answer me at that moment (it's the first time he works teaching about rings). The question goes like this:

Consider $R$ a ring. Being $I\leq R$ a subring of $R$, which of the following statements is true?

1. $R$ field $\implies$ $I$ field.
2. $R$ euclidean domain $\implies$ $I$ euclidean domain.
3. $R$ principal ideal domain $\implies$ $I$ principal ideal domain.
4. $R$ Unique factorisation domain $\implies$ $I$ Unique factorisation domain.
5. $R$ integral domain domain $\implies$ $I$ integral domain.

I don't know if these are just dumb questions, I'm a bit new to abstract algebra and all these definitions of rings tend to confuse me. Is any of the above statements true? Is any of them true for $I$ ideal subring of $R$? Can you give counterexamples for those false? Is any true is I use the word "ring" instead of "domain"?

Any help will be appreciated, thanks in advance.