Abstract
We investigate the relationship between various choice principles and nhbox {th}-root functions in rings. For example, we show that the Axiom of Choice is equivalent to the statement that every ring has a square-root function. Furthermore, we introduce a choice principle which implies that every integral domain has an nhbox {th}-root function (for odd integers n), and introduce another choice principle which is equivalent to the Prime Ideal Theorem restricted to certain ideals. Finally, we investigate the dependencies between the two new choice principles and a choice principle for families of n-element sets.
Highlights
Some Forms of Choice Related to Algebra The investigation of consequences of the Axiom of Choice in algebra has a long tradition
This problem was solved by Hodges [4], who showed that Krull’s Theorem is equivalent to the Well-Ordering Principle, which is in turn equivalent to the Axiom of Choice
Prime Ideal Theorem (FORM 14 C in [5]): Every commutative ring with a unit has a prime ideal. This choice principle is weaker than the Axiom of Choice and is for example equivalent to the following statement: For every graph G, if every finite subgraph of G is 3-colorable, G is n-colorable, n ≥ 3
Summary
Some Forms of Choice Related to Algebra The investigation of consequences of the Axiom of Choice in algebra has a long tradition. We list a few choice principles in the context of rings and vector spaces. For more choice principles related to rings we refer the reader to Howard and Rubin [5, pp. Krull’s Theorem (FORM 1 CD in [5]): Every proper ideal in a commutative ring can be extended to a maximal ideal. Krull proved in [7] that every non-zero ring has a maximal ideal. Since he used explicitly the Well-Ordering Principle (FORM 1 CD in [5]) in his proof Since he used explicitly the Well-Ordering Principle (FORM 1 CD in [5]) in his proof (see [7, p. 735 f]), one may ask how much of the Axiom of Choice we get Salome Schumacher: Partially supported by SNF grant 200021 178851
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