# Centre of the ring of quaternions

by Wingeer
Tags: centre, quaternions, ring
 P: 79 1. The problem statement, all variables and given/known data What is the centre of the ring of the quaternions defined by: $$\mathbf{H}=\{ \begin{pmatrix} a & b \\ -\bar{b} & \bar{a} \end{pmatrix} | a,b \in \mathbf{C} \}$$? 2. Relevant equations The definition of the centre of a ring: The centre Z of a ring R is defined by $$Z(R)=\{A | AX=XA, \forall X \in R\}$$ 3. The attempt at a solution I figured that multiples of the 2x2 identity matrix must be in the centre. Also if we denote an element of H by: $$\begin{pmatrix} x & y \\ -\bar{y} & \bar{x} \end{pmatrix}$$ where $$x=x_1 + ix_2$$ and similarly for a,b and y that: 1. $$b\bar{y}=\bar{b}y$$ 2. $$y(a-\bar{a})=b(x-\bar{x})$$ 3. $$\bar{b}(x-\bar{x})=\bar{y}(a-\bar{a})$$ Then for instance we get from the first equation that: $$b_2x_1=a_1y_2$$ But I am not sure whether this approach really is any useful at all. Some hints would be greatly appreciated.
 P: 79 Anyone? I actually have another question about the quaternions. I am asked to show that: $$\mathbf{H'} = \{ a+bi+cj+dk | a,b,c,d \in \mathbf{R} \}$$ with: i^2=j^2=k^2=-1, ij=k=-ji, ik=-j=-ki and jk=i=-kj. is isomorphic as rings to the quaternions defined in the previous post. I started by noticing that (where x,y are complex numbers): $$\begin{pmatrix} x & y \\ -\bar{x} & \bar{y} \end{pmatrix} = \begin{pmatrix} a+bi & c+di \\ c-di & a-bi \end{pmatrix} = \begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix} + \begin{pmatrix} bi & 0 \\ 0 & bi \end{pmatrix} + \begin{pmatrix} 0 & c \\ -c & 0 \end{pmatrix} + \begin{pmatrix} 0 & di \\ di & 0 \end{pmatrix}$$ And so we see that every element in H is a linear combination of these matrices which all are linearly independent as well. This means we have found a basis for H. So if we define a function f:H -> H' by: $$1= f \left( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right)$$ $$i=f \left( \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}\right)$$ $$j=f \left( \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\right)$$ $$k=f \left( \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}\right)$$ We see that obviously f is both surjective and injective as these are the only values f are defined for. Therefore f is an bijection and H and H' are isomorphic. Do I have to mix ring homomorphisms in this? Or?