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- Jun 22, 2012

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

Let K be any ring with [TEX] 1 \in K [/TEX] whose center is a field and [TEX] 0 \ne x, 0 \ne y \in [/TEX] 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:

[TEX] I^2 = x, J^2 = y, IJ= -JI, cI = Ic, cJ = Jc, cIJ = IJc [/TEX] for all [TEX] c \in K [/TEX]

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

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I am somewhat overwhelmed by this problem and its notation.

Can someone please help me get started?

Peter

[This has also been posted on MHF]

Let K be any ring with [TEX] 1 \in K [/TEX] whose center is a field and [TEX] 0 \ne x, 0 \ne y \in [/TEX] 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:

[TEX] I^2 = x, J^2 = y, IJ= -JI, cI = Ic, cJ = Jc, cIJ = IJc [/TEX] for all [TEX] c \in K [/TEX]

(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

[This has also been posted on MHF]

Last edited: