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

[Math Amateur]

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

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

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

(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]

 

[Deveno]

The basic idea is to generalize the concept of "ordinary" quaternions (where \(\displaystyle K = \Bbb R\) and \(\displaystyle x = y = -1\)) to a more abstract setting (for example, K might be a finite field, and I and J might be obtained from extension fields of K via quotients of K[x]).

K-linearity is going to give you the abelian group structure, so that's a non-issue. Multiplicative associativity and the distributive axioms of a ring are where you should concentrate your efforts.

 

[Math Amateur]

Thanks Deveno

Will have another try at this problem this evening (AEST - Australian Eastern Standard Time)

Peter

 

[Math Amateur]

Thanks Deveno

Will have another try at this problem this evening (AEST - Australian Eastern Standard Time)

Peter

I think I need more help ...

Initial 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

 

[Math Amateur]

I think I need more help ...

Initial 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

Again, reflecting on this problem I went to Dummit and Foote Chapter 7 and checked their description of (real) Hamiltonian Quaternions on the bottom of page 224 and the top of page 225 (see attached). There they define the operations of addition and multiplication (along with some relations among i , j, and k for simplification)

If one accepts these definitions and applies the operations so defined to Dauns exercise where the elements a, b, c and so on belong not to the real numbers but to a general ring K.

The proof that K is an associative ring is not difficult but is tedious.

I have one problem left - in verifying K[I, J] was a ring I did not seem to use the fact that K's center is a field. Can someone please indicate to me where this is needed?

Peter

 

[Deveno]

To prove the distributive law, at some point you will need to use the fact that x and y commute.

The fact that Z(K) is a field (and not just a commutative ring) is used to show that K[I,J] is a division ring, and not merely an associative ring.

One can think of K[I,J] as being a quotient of a polynomial ring:

\(\displaystyle K[I,J] = K[X,Y]/\langle X^2-x,Y^2-y,XY +YX\rangle\)

in other words we adjoin a root of \(\displaystyle X^2 - x\) which we call I, and similarly for J, and ensure that I and J anti-commute.

 

[Math Amateur]

To prove the distributive law, at some point you will need to use the fact that x and y commute.

The fact that Z(K) is a field (and not just a commutative ring) is used to show that K[I,J] is a division ring, and not merely an associative ring.

One can think of K[I,J] as being a quotient of a polynomial ring:

\(\displaystyle K[I,J] = K[X,Y]/\langle X^2-x,Y^2-y,XY +YX\rangle\)

in other words we adjoin a root of \(\displaystyle X^2 - x\) which we call I, and similarly for J, and ensure that I and J anti-commute.

Thaanks Deveno

I will rework the distributive laws more carefully

Still reflecting over the other (very interesting) points you make ... Will do some work on these points

J Thanks again,

Peter