I'm working on a problem from Pinter's Abstract Algebra and am wondering if someone can tell me if I'm on the right track.
Let $A$ be a finite integral domain. Prove that if there are distinct nonzero elements $a$ and $b$ in $A$ such that $125 \cdot a = 125 \cdot b$, then $A$ has characteristic 5.
As this is a finite integral domain, it has nonzero characteristic. By Theorem 20.2, that characteristic must be a prime number. By Theorem 20.1, all nonzero elements in an integral domain have the same additive order $n$. So, unity (1) has additive order $n$, as do $a,b \in A$.
Because $a,b$ have order $n$, $n \cdot a = 0$ and $n \cdot b = 0$. This implies that $n \cdot a = n \cdot b$. And since $125 \cdot a = 125 \cdot b$, and according to Theorem 10.5, 125 must be a multiple of $n$. The only prime factor of 125 is 5, so $char(A) = 5$.
Theorem 10.5: Suppose an element $a$ in a group has order $n$. Then $a^t = e$ iff $t$ is a multiple of $n$.
Theorem 20.1: All nonzero elements in an integral domain have the same additive order.
Theorem 20.2: In an integral domain with nonzero characteristic, the characteristic is a prime number.