I'm trying to solve a problem which asks to find how many elements the multiplicative group of a ring $R$ has. It is defined as follows: $R= \Big\{ \left(\begin{matrix} a & b \\ -b & a \\ \end{matrix}\right) | a,b \in \Bbb Z_3\Big\}$
To look for how many elements the multiplicative group has is not the same as to look for how many elements the general linear group has? Anyway, I know that with $\Bbb Z_n$ it is possible to use the Euler function to know how many elements the corresponding multiplicative group has. How should I proceed in this case? Any help?
There are only nine element of this ring, and the invertible ones satisfy $a^2+b^2\neq 0$. So write out the matrices and check the condition (You will find that there are 8 such matrices).