I'm new to this topic, so any help would be appriciated!

Note: these are all in a communitive ring $\displaystyle R $.

1. $\displaystyle a \mid b \Longleftrightarrow (b) \subseteq (a) $.

2. $\displaystyle u \in R $ is a unit $\displaystyle \Longleftrightarrow (u) = R $.

3. $\displaystyle a $ and $\displaystyle b $ are associative $\displaystyle \Longleftrightarrow (a) = (b) $.

4. $\displaystyle p $ is prime $\displaystyle \Longleftrightarrow ab \in (p) $ implies that either $\displaystyle a \in (p) $ or $\displaystyle b \in (p) $.