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Divisors of finite character. (English) Zbl 0533.20034

The topic of this very extensive paper (35 pages) is the theory of divisibility which goes back to Kummer’s ”ideal numbers”. This theory can be investigated in a purely multiplicative form. In this form Krull (1924) began to develop this theory. Arnold (1929) and Clifford (1938) were interested in special semigroups like multiplicative semigroups of Dedekind domains. Lorenzen (1939), Jaffard (1960) in his monograph and Aubert (1962) developed the general theory of various systems of ideals for multiplicative structures.
The important notion in this direction is shown to be the ”t-ideal” which is defined as follows. We put for a bounded subset A of a directed ordered abelian group G \(A_ v=G^+:(G^+:A)\quad(X:Y=\{g\in G;\quad g.Y\subseteq X\})\) and the t-ideal generated by A is the set \(A_ t=\cup N_ v\quad(N\subseteq A,\quad N\quad finite).\)
This paper is devoted to the notion of t-ideal in various directions of investigation. Let us add the author’s summary: ”The present paper purports to show that divisors of finite character - also called t-ideals - are the natural building blocks of the general theory of divisibility. Divisors of finite character are here applied to a variety of different arithmetical topics as well as to sectional and functional representations of ordered groups.”
Reviewer: L.Skula

MSC:

20M12 Ideal theory for semigroups
16Dxx Modules, bimodules and ideals in associative algebras
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
13A15 Ideals and multiplicative ideal theory in commutative rings
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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References:

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