Question 66
Let $(A,+,\cdot)$ be a ring satisfying both conditions:
(1) $A$ is not a field;
(2) for any noninvertible element $x$ of $A$, there is an integer $m\ge1$, depending on $x$, such that
$$x=x^2+x^3+\cdots+x^{2^m}.$$
Show that:
a. $x+x=0$ for every $x\in A$;
b. $x^2=x$ for every noninvertible element $x\in A$.
(1) $A$ is not a field;
(2) for any noninvertible element $x$ of $A$, there is an integer $m\ge1$, depending on $x$, such that
$$x=x^2+x^3+\cdots+x^{2^m}.$$
Show that:
a. $x+x=0$ for every $x\in A$;
b. $x^2=x$ for every noninvertible element $x\in A$.