In Dauns book "Modules and Rings", Exercise 19 in Section 1-5 reads as follows: (see attachment)
Let K be any ring with 1∈K whose center is a field and 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!)
for all
Prove that the ring K[I, J] is isomorphic to a ring of matrices as follows:
for all
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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