If F is a field prove that the principal ideal x is maximal

If F is a field, prove that the principal ideal (x) is maximal in the ring F[x]

Solution

If F is a filed, then the only maximal ideal is {0}.

All non-zero prime ideals are maximal in a principal ideal domain.

Let R be the principal ideal domain.

A principal ideal domain R is an integral domain in which every ideal A is of the form <A>={ar/r in R}

Let <a> be a prime ideal

R/A is an integral domain

R/A is finite

Then R/A is a field.

Then A is a maximal ideal of F