Question 35
Let $(A, +, \cdot)$ be a unitary ring such that the following are true:
(1) $A$ is not a field;
(2) for any noninvertible element $x$ of $A$, there is an integer $m \ge 1$, depending on $x$, such that $x = x^2 + x^3 + \dots + x^{2^m}$. Prove that:
a. $x+x=0$, for any $x \in A$,
b. $x^2 = x$, for any noninvertible $1 \neq x \in A$.
(1) $A$ is not a field;
(2) for any noninvertible element $x$ of $A$, there is an integer $m \ge 1$, depending on $x$, such that $x = x^2 + x^3 + \dots + x^{2^m}$. Prove that:
a. $x+x=0$, for any $x \in A$,
b. $x^2 = x$, for any noninvertible $1 \neq x \in A$.