Proof: Let F be the set of all subrings of R not containing 12, and let C be a typical chain of subrings R0⊆R1⊆.... AdmitR=⋃n∈NRnand prove it is an upper bound of C. For any a,b∈R, we have a∈Rx and b∈Ry for some x,y implying a−b∈R and ab∈R so that R is a subring. Furthermore, by definition of the union, 1∈R and 12∉R. By Zorn's Lemma, F has a maximal element. ◻
Thursday, May 2, 2013
Zorn's Lemma on the Real Numbers (7.5.6)
Dummit and Foote Abstract Algebra, section 7.5, exercise 6:
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Prove that R contains a subring A with identity and maximal under inclusion such that 12∉A.
Proof: Let F be the set of all subrings of R not containing 12, and let C be a typical chain of subrings R0⊆R1⊆.... AdmitR=⋃n∈NRnand prove it is an upper bound of C. For any a,b∈R, we have a∈Rx and b∈Ry for some x,y implying a−b∈R and ab∈R so that R is a subring. Furthermore, by definition of the union, 1∈R and 12∉R. By Zorn's Lemma, F has a maximal element. ◻
Proof: Let F be the set of all subrings of R not containing 12, and let C be a typical chain of subrings R0⊆R1⊆.... AdmitR=⋃n∈NRnand prove it is an upper bound of C. For any a,b∈R, we have a∈Rx and b∈Ry for some x,y implying a−b∈R and ab∈R so that R is a subring. Furthermore, by definition of the union, 1∈R and 12∉R. By Zorn's Lemma, F has a maximal element. ◻
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