Question 99
Let $k \in \mathbb{N}^*$. We say that the ring $(A, +, \cdot)$ has the property $CP(k)$, if for every $a, b \in A$ there is a $c \in A$, such that $a^k = b^k + c^k$.
a) Give an example of a finite ring $(A, +, \cdot)$, which does not have the property $CP(k)$ for any positive integer $k$, with $k \ge 2$.
b) Let $n \in \mathbb{N}$, $n \ge 3$, and $M(n) = \{m \in \mathbb{N}^* \mid (\mathbb{Z}_n, +, \cdot) \text{ has the property } CP(m)\}$. Prove that $M(n)$ is a monoid with respect to multiplication, included in the set $2 \cdot \mathbb{N} + 1$ of odd positive integers.
a) Give an example of a finite ring $(A, +, \cdot)$, which does not have the property $CP(k)$ for any positive integer $k$, with $k \ge 2$.
b) Let $n \in \mathbb{N}$, $n \ge 3$, and $M(n) = \{m \in \mathbb{N}^* \mid (\mathbb{Z}_n, +, \cdot) \text{ has the property } CP(m)\}$. Prove that $M(n)$ is a monoid with respect to multiplication, included in the set $2 \cdot \mathbb{N} + 1$ of odd positive integers.