How can a prime element in a Ring be proven as irreducible?

[NoodleDurh]

How does one show that a prime element in a Ring is irreducible and how does one show that $$ iff x is a unit.

okay from my knowledge I know that units are invertible elements, so how does the norm of x make it 1... maybe I am not too sure about this

 

[Office_Shredder]

For your first question, try doing a proof by contradiction. Let p be a prime element, and suppose it is reducible. What can you do with that?