pm-Rings and the Prime Ideal Theorem |
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Authors: | B. Banaschewski |
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Affiliation: | Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada |
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Abstract: | A commutative ring A with unit is called a pm-ring if every prime ideal of A is contained in a unique maximal ideal, and a Gelfand ring if a+b=1 in A implies that (1+ar)(1+bs)=0 for some r,s∈A. It was shown earlier, in a somewhat circuitous way involving pointfree topology, that “pm implies Gelfand” iff the Prime Ideal Theorem holds. The present note provides an alternative, more direct and entirely ring theoretical proof of a somewhat augmented version of this result. |
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Keywords: | Gelfand ring pm-Ring Prime Ideal Theorem |
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