Is Every Ideal Being Prime Indicative of a Commutative Ring Being a Field?

[fireisland27]

Homework Statement

Given a commutative ring with unity, show that if every ideal is prime than the ring is a field.

Homework Equations

The Attempt at a Solution

I think that I can show that a ring is a field iff it has no nontrivial ideals. So I guess I need to show that if a ring has only prime ideals than these ideals must be trivial. I'm not sure how to do this though.

 

[Petek]

Can you show that the ring must be an integral domain? (Hint: Consider the ideal that consists just of the zero element.) Next show that every non-zero element is a unit to complete the proof. (Hint: Let r be a non-zero element. Consider the ideal generated by $r^2$.)

Petek