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In Dauns book "Modules and Rings", Exercise 19 in Section 1-5 reads as follows: (see attachment)
Let K be any ring with [FONT=MathJax_Main]1[FONT=MathJax_Main]∈[FONT=MathJax_Math]K 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 $$ a,b \in K $$
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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
Let K be any ring with [FONT=MathJax_Main]1[FONT=MathJax_Main]∈[FONT=MathJax_Math]K 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 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
Last edited: