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

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    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  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!)

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

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

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

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

    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
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