Prime ideals and fields

[j18eos]

This exercise isn't my, however I write the source: Kaplansky Commutative Rings (2nd edition), exercise page 7 number 1!

Let \(R\) be a ring. Suppose that every ideal of \(R\) (other than \(R\)) is prime. Prove that \(R\) is a field.

You can use:

• the definition of prime ideal;
• the (saturated) multiplicatively closed sets;
• the theory of UFD and PID;
• the theorems of Herstein and Cohen about prime ideals;
• the definition of noetherian ring.

Good work!

[j18eos]

Testo nascosto, fai click qui per vederlo

Prove that \(R\) is a U.F.D.!