91Ó°ÊÓ

Show that a commutative ring \(R\) is a field if and only if 0 is the only proper ideal of \(R\).

vLet \(R\) be a commutative ring. An ideal \(I\) of \(R\) is said to be prime if the following holds. P: If \(r, s \in R\) and \(r s \in I\), then either \(r \in I\) or \(s \in I\) (possibly both are in \(I\) ). Show that (i) 0 is a prime ideal of \(R \Longleftrightarrow R\) is a domain. (ii) \(I\) is prime \(\Longleftrightarrow R / I\) is a domain. (iii) A maximal ideal of \(R\) must be a prime ideal.