An exercise on a ring

Let $\mathcal{R}$ be a ring and we suppose that $a, b \in \mathcal{R}$ be two elements of $\mathcal{R}$ such that $ab=1$. Prove that:

1. $a^n b^n =1$ forall $n \geq 1$.
2. $(1-ba)b^n =0$ forall $n \geq 1$.
3. the elements $ba$ and $1-ba$ are idempotent and orthogonal.
4. if $c \in \mathcal{R}$ and $ca=1$ , then the element $a$ is invertible and it holds that:
   
   $$a^{-1}=b=c$$

Solution

 

1. Let us use induction on this one. For $n=1$ the result obviously holds by the assumptions. Let $n \geq 2$ and assume that $a^n b^n =1$ Thus:
   
   \begin{align*}
   a^{n+1}b^{n+1} &= a^n a b b^n\\
    &=a^n b^n \\
    &=1
   \end{align*}


2. We have successively:
   
   \begin{align*}
   \left ( 1-ba \right )b^n  &=b^n -ba b^n  \\
    &=b^n -bab b^{n-1} \\
    &= b^n - b b^{n-1}\\
    &= b^n - b^n \\
    &=0
   \end{align*}


3. We have that:
   
   1. \begin{align*} \left ( ba \right )^2 &=ba ba \\ &= ba\\ \end{align*}
   2. \begin{align*}
      \left ( 1-ba \right )^2 &= \left ( 1-ba \right )\left ( 1-ba \right )\\
       &= 1-ba -ba + ba ba\\
       &= 1-ba-ba + ba\\
       &= 1-ba
      \end{align*} 
   That these two elements are orthogonal is immediate if we multiply them.
   


4. We have $ca=1 \Rightarrow cab=b \Rightarrow c=b \Rightarrow ab=1=ba$.Hence $a$ is invertible with inverse $b$ and $c$.