Question 36
Let $(A, +, \cdot)$ be a unit ring with the property: for all $x \in A$,
$$
x + x^2 + x^3 = x^4 + x^5 + x^6.
$$
a) Let $x \in A$ and let $n \ge 2$ be an integer such that $x^n = 0$. Prove that $x = 0$.
b) Prove that $x^4 = x$, for all $x \in A$.
$$
x + x^2 + x^3 = x^4 + x^5 + x^6.
$$
a) Let $x \in A$ and let $n \ge 2$ be an integer such that $x^n = 0$. Prove that $x = 0$.
b) Prove that $x^4 = x$, for all $x \in A$.