Sentence:  In the rings studied in Section 17.4 a nonzero <WSD>prime ideal</WSD> is divisible only by itself and by \mathfrak{o} on the basis of Axiom II; thus, in that section there are no lower <WSD>prime ideals</WSD> but \mathfrak{o}. Since every ideal \mathfrak{a}\ne\mathfrak{o} is divisible by a <WSD>prime ideal</WSD> distinct from \mathfrak{o} (proof: from among all the divisors of a distinct from \mathfrak{o} choose a maximal one; since this ideal is maximal it is also prime), it follows that a cannot be quasi-equal to \mathfrak{o}. 
Meaning of the word marked with <WSD> tags:  Any (two-sided) ideal I such that for arbitrary ideals P and Q, PQ\subseteq I\implies P\subseteq I or Q\subseteq I.