# Generalised Quaternion Algebra over K - Dauns Section 1-5 no 17

• Aug 12th 2013, 03:38 PM
Bernhard
Generalised Quaternion Algebra over K - Dauns Section 1-5 no 17
In Dauns book "Modules and Rings", Exercise 17 in Section 1-5 reads as follows: (see attachment)

Let K be any ring with $\displaystyle 1 \in K$ whose center is a field and $\displaystyle 0 \ne x, 0 \ne y \in$ 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:

$\displaystyle I^2 = x, J^2 = y, IJ= -JI, cI = Ic, cJ = Jc, cIJ = IJc$ for all $\displaystyle c \in K$

(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.

Peter
• Aug 13th 2013, 12:49 AM
Bernhard
Re: Generalised Quaternion Algebra over K - Dauns Section 1-5 no 17
Quote:

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

Let K be any ring with $\displaystyle 1 \in K$ whose center is a field and $\displaystyle 0 \ne x, 0 \ne y \in$ 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:

$\displaystyle I^2 = x, J^2 = y, IJ= -JI, cI = Ic, cJ = Jc, cIJ = IJc$ for all $\displaystyle c \in K$

(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.

Peter

I have now reflected on this problem and received some help from Deveno.

However I still need some help.

My further thoughts are as follows:

K is a ring with 1. Center of ring K is a field

I, J and IJ are symbols not in K (???)

The set K[I, J] = K + KI + KJ = KIJ is the set of all K-linear combinations of {1, I , J, IJ}

So now let $\displaystyle a, b \in K$ and let

$\displaystyle X = a + aI + aJ + aIJ \in K[I, J]$

and

$\displaystyle Y = b + bI + bJ + bIJ \in K[I, J]$

But if K[I, J] is an associative ring then we must have X+Y = Y + X

ie we need

$\displaystyle a + aI + aJ + aIJ + b + bI + bJ + bIJ = b + bI + bJ + bIJ + a + aI + aJ + aIJ$

But ???

How do we manipulate these expressions ie how do we manipulate the the aI, bI, aJ, bJ, ... etc

We know how to manipulate field or ring elements but these are field elements 'times' symbols not in K

Can anyone clarify this situation for me

Peter