Thread: Generalised Quaternion Algebra over K - Dauns Section 1-5 no 19

1. Generalised Quaternion Algebra over K - Dauns Section 1-5 no 19

In Dauns book "Modules and Rings", Exercise 19 in Section 1-5 reads as follows: (see attachment)

Let K be any ring with 1K 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}.

The following multiplication rules apply: (These also apply in my post re Ex 18!)

$\displaystyle I^2 = x, J^2 = y, IJ = -JI, cI = Ic, cIJ = JIc$ for all $\displaystyle c \in K$

Prove that the ring K[I, J] is isomorphic to a ring of $\displaystyle 2 \times 2$ matrices as follows:

$\displaystyle a + bJ \rightarrow \begin{pmatrix} a & by \\ \overline{b} & \overline{a} \end{pmatrix}$ for all $\displaystyle a,b \in K[I]$

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I am not sure how to go about this ... indeed I am confused by the statement of the problem. My issue is the following:

Elements of K[I, J] are of the form r = a + bI + cJ + dIJ so we would expect an isomorphism of K[I, J] to specify how elements of this form are mapped into another ring, but we are only told how elements of the form s = a + bJ are mapped. ???

Can someone please clarify this issue and help me to get started on this exercise?

Peter