Exercise 1, § 4

(a) Let G be a totally ordered group and M a major set in G, not containing 0. Show that there exists a greatest isolated subgroup H of G not meeting M.

(b) Let A be a valuation ring and $v$ a valuation associated with A. For an ideal $\varrho \neq 0$ of A to be prime, it is necessary and sufficient that p correspond to a major set M in the totally ordered value group G of $v$ such that, if H is the greatest isolated subgroup of G not meeting M, M is the complement of $H_+$ in G,.

(c) With the notation of (b), for an ideal q of A to be p-primary, it is necessary and sufficient that it satisfy one of the following conditions: either $H = \{0\}$ (and p is then maximal) or $q = \varrho$.

(d) For an ideal a of A which is distinct from 0 and A to satisfy $a^2 = a$, it is necessary and sufficient that a correspond to a major set M such that, if H is the greatest isolated subgroup of G not meeting M, the image $M'$ of M in the totally ordered group $G' = G/H$ has no least element; a is then a prime ideal.