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Nilradicals and Prime Ideals

Assume R is commutative ring. Prove that the following are equivalent:
a) R has exactly 1 prime ideal
b) Every element of R is either nilpotent or a unit.
c) R/N(R) is a field

Here N(R) denotes the nilradical of R.
The set of nilpotent elements form an ideal called the nilradical of R

Solution Summary

Nilradicals and Prime Ideals are investigated.