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Thursday, May 2, 2013

Zorn's Lemma on the Real Numbers (7.5.6)

Dummit and Foote Abstract Algebra, section 7.5, exercise 6:

MathJax TeX Test Page Prove that R contains a subring A with identity and maximal under inclusion such that 12A.

Proof: Let F be the set of all subrings of R not containing 12, and let C be a typical chain of subrings R0R1.... AdmitR=nNRnand prove it is an upper bound of C. For any a,bR, we have aRx and bRy for some x,y implying abR and abR so that R is a subring. Furthermore, by definition of the union, 1R and 12R. By Zorn's Lemma, F has a maximal element. 

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