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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Chain conditions in computable rings


Author: Chris J. Conidis
Journal: Trans. Amer. Math. Soc. 362 (2010), 6523-6550
MSC (2000): Primary 03B30, 03D80; Secondary 03F35
Published electronically: July 15, 2010
MathSciNet review: 2678985
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Abstract: Friedman, Simpson, and Smith showed that, over RCA$ _0$, the statements ``Every ring has a maximal ideal'' and ``Every ring has a prime ideal'' are equivalent to ACA$ _0$ and WKL$ _0$, respectively. More recently, Downey, Lempp, and Mileti have shown that, over RCA$ _0$, the statement ``Every ring that is not a field contains a nontrivial ideal'' is equivalent to WKL$ _0$.

In this article we explore the reverse mathematical strength of the classic theorems from commutative algebra which say that every Artinian ring is Noetherian, and every Artinian ring is of finite length. In particular we show that, over RCA$ _0$, the former implies WKL$ _0$ and is implied by ACA$ _0$, while over RCA$ _0$+B$ \Sigma_2$, the latter is equivalent to ACA$ _0$.


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Additional Information

Chris J. Conidis
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Address at time of publication: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: conidis@math.uchicago.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05013-5
PII: S 0002-9947(2010)05013-5
Keywords: Mathematical logic, computability theory, reverse mathematics, ring theory
Received by editor(s): December 1, 2008
Published electronically: July 15, 2010
Additional Notes: The author was partially supported by NSERC grant PGS D2-344244-2007. Furthermore, he would like to acknowledge the helpful input he received from his thesis advisors, R.I. Soare, D.R. Hirschfeldt, and A. Montalbán. He would also like to especially thank J.R.\ Mileti for suggesting this topic, and A. Montalbán for his help in editing the initial drafts.
Dedicated: This paper is dedicated to the author’s thesis advisors: Robert I. Soare, Denis R. Hirschfeldt, and Antonio Montalbán.
Article copyright: © Copyright 2010 American Mathematical Society