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

Let K be any ring with 1∈Kwhose 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[I] = 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[I] $. Show that $\displaystyle \overline{q} = \overline{a} + \overline{b}J $.

---------------------------------------------------------------------

I can show that K[I] = 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 MHB]