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Minimal prime ideals and arithmetic comprehension

Published online by Cambridge University Press:  12 March 2014

Kostas Hatzikiriakou
Kostas Hatzikiriakou*
Affiliation:
Pavlou Mela 42, Thessaloniki 54622, Greece
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Extract

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 1 , March 1991 , pp. 67 - 70
Copyright
Copyright © Association for Symbolic Logic 1991

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References

REFERENCES

[1]Friedman, H., Simpson, S. G., and Smith, R. L., Countable algebra and set existence axioms, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 141–181; addendum, vol. 27 (1983), pp. 319–320.CrossRefGoogle Scholar
[2]Kaplansky, I., Commutative rings, The University of Chicago Press, Chicago, Illinois, 1974.Google Scholar
[3]Simpson, S. G., Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations? this Journal, vol. 49 (1984), pp. 783–802.Google Scholar