Abstract
It is shown that the axiom of choice AC is equivalent to the statements: (1)For every Boolean ring (B, +, ·)and every subset H ⊑B which is closed under + there exists a ⊑−maximal ideal Q⊑B such that H ∩Q={0} and, (2)For every Boolean ring (B,+,·),for every A ⊑B, the infinite system:X i + yi=bi,iεI, b i ε B has a solution in A iff each of its equations has a solution in A.
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References
Armbrust, M. K.,An algebraic equivalent of a multiple choice axiom, Fund. Math.74 (1972), 145–146.
Blass, A.,Existence of bases implies the axiom of choice, in Axiomatic Set Theory, Contemporary Mathematics31 (1984), 31–33.
Bleicher, M.,Some theorems on vector spaces and the axiom of choice, Fund. Math.54 (1964), 95–107.
Howard, P. E.,Some strong forms of the Boolean prime ideal theorem, Notices A. M. S.24 (1977), A446.
Jech, T.,The Axiom of Choice, North-Holland, 1973.
Keremedis, K.,Bases for vector spaces over the two element field and AC, Proc. Amer. Math. Soc.124 (1996), 2527–2531.
Mrowka, S.,Two remarks on my paper “On the ideals extension theorem and its equivalence to the axiom of choice”, Fund. Mat.46 (1958), 163–166.
Rubin, H. andRubin, J. E.,Equivalents of the Axiom of Choice, II, North-Holland, 1985.
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Keremedis, K. Some equivalents of AC in algebra. Algebra Universalis 36, 564–572 (1996). https://doi.org/10.1007/BF01233925
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DOI: https://doi.org/10.1007/BF01233925