In Dauns book "Modules and Rings", Exercise 17 in Section 1-5 reads as follows: (see attachment)

Let K be any ring with

whose center is a field and

center K 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}.

Show that K[I,J] becomes an associative ring under the following multiplication rules:

for all

(K[I, J] is called a generalised quaternion algebra over K)

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

I am somewhat overwhelmed by this problem and its notation.

Can someone please help me get started?

Peter