Semi-Valuations and Groups of Divisibility

Abstract

Associated with any integral domain R there is a partially ordered group A , called the group of divisibility of R . When R is a valuation ring, A is merely the value group; and in this case, ideal-theoretic properties of R are easily derived from corresponding properties of A , and conversely. Even in the general case, though, it has proved useful on occasion to phrase a ring-theoretic problem in terms of the ordered group A , first solve the problem there, and then pull back the solution if possible to R. Lorenzen ( 15 ) originally applied this technique to solve a problem of Krull, and Nakayama ( 16 ) used it to produce a counterexample to another question of Krull. More recently, Heinzer ( 7;8 ) has used the method to construct other interesting examples of rings.