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Semi-Valuations and Groups of Divisibility

Published online by Cambridge University Press:  20 November 2018

Jack Ohm
Jack Ohm*
Affiliation:
Louisiana State University, Baton Rouge, Louisiana
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 576 - 591
Copyright
Copyright © Canadian Mathematical Society 1969

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