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- Jun 22, 2012

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In Dauns book "Modules and Rings", Exercise 18 in Section 1-5 reads as follows: (see attachment)

Let K be any ring with [FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Math]

Let I, J, and IJ be symbols not in K.

Form the set K[I, J] = K + KI + KJ + KIJ of all K linear combinations of {1, I, J, IJ}.

Prove that the subring K

Let K be any ring with [FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Math]

*K*[/FONT] whose center is a field and \(\displaystyle 0 \ne x, 0 \ne y \in \) center K are any elements.Let I, J, and IJ be symbols not in K.

Form the set K[I, J] = K + KI + KJ + KIJ of all K linear combinations of {1, I, J, IJ}.

Prove that the subring K

*= K + KI has an automorphism of order 2 defined by \(\displaystyle a = a_1 + a_2I \rightarrow \overline{a} = a_1 - a_2I \) for \(\displaystyle a_1, a_2 \in K \) which extends to an inner automorphism of order two of all of K[I,J], where \(\displaystyle q \rightarrow \overline{q} = J_{-1}qJ = (1/y)JqJ \) for \(\displaystyle q = a + bJ, a, b \in K \)**. Show that \(\displaystyle \overline{q} = \overline{a} + \overline{b}J \).*

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I can show that K---------------------------------------------------------------------

I can show that K

*= K + KI is an automorphism, but I am unsure of the meaning of an automorphism having order two - let alone proving it! So I would appreciate some help on showing that the automorphism has order two.*

I cannot however show that it 'extends' to the inner automorphism of order two that is then defined.

I would appreciate help with these parts of the exercise.

Peter

[This has also been posted on MHF]I cannot however show that it 'extends' to the inner automorphism of order two that is then defined.

I would appreciate help with these parts of the exercise.

Peter

[This has also been posted on MHF]

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