Question 12
a) Prove that, if a ring $(A, +, \cdot)$ has property (P) and $a, b$ are distinct elements of $A$ such that $a$ and $a+b$ are invertible, then $b$ is not invertible, but $1+ab$ is invertible.
b) Give an example of a unitary ring possessing (P).
where property (P) is:
$$
(P) \quad \left\{ \begin{array}{l} \text{the set } A \text{ has at least 4 elements,} \\ \text{the element } 1+1 \text{ is invertible in } A, \\ x+x^4 = x^2+x^3, \text{ for any } x \in A. \end{array} \right.
$$
b) Give an example of a unitary ring possessing (P).
where property (P) is:
$$
(P) \quad \left\{ \begin{array}{l} \text{the set } A \text{ has at least 4 elements,} \\ \text{the element } 1+1 \text{ is invertible in } A, \\ x+x^4 = x^2+x^3, \text{ for any } x \in A. \end{array} \right.
$$