First problem: I have set of matrices \begin{pmatrix} x & y \\ ry & x \end{pmatrix} where, $x,y \in R$, $R$ is a ring, and $r$ is a fixed element from $R$. I need to proof that this set is a ring with respect to matrix multiplication and addition.
My attempt:
- I need to show that it is an abelian group with respect to addition.
1.1) Associativity is quite simple
1.2)I need to determine the Identity element. The identity element here is $$\begin{pmatrix} x & y \\ ry & x \end{pmatrix} + \begin{pmatrix} e & e \\ re & e \end{pmatrix}= \begin{pmatrix} x & y \\ ry & x \end{pmatrix} $$
where $e$ is identity element from $R$
1.3) Need to determine the inverse element
$$\begin{pmatrix} x & y \\ ry & x \end{pmatrix} + \begin{pmatrix} -x & -y \\ r(e-y) & -x \end{pmatrix} = \begin{pmatrix} e & e \\ re & e \end{pmatrix}$$ 1.4) the group with respect to + is abelian
- To proof that multiplication is distributive with respect to addition we need just doing matrix multiplication, nothing special here.
Am I missing something in the first case?
Second problem: I have another set of matrices: $$\frac{1}{2} \begin{pmatrix} x & y \\ ay & x \end{pmatrix}$$
where $a \in Z$ and not divisible by square of prime, and $x,y \in Z$ have the same parity. How these conditions effect the proof? I don't see any differences with proof if $x,y,a \in Z$.