Context: Let R be a unital ring. The characteristic of R is the smallest positive integer n such that n⋅1=0. If no such n exists, we say R has characteristic 0. We denote the characteristic of a ring by char(R).
Question: Every unital ring of characteristic zero is infinite (I'm thinking of using a proof by contradiction for this, but I have no idea how).
Could you provide an answer that avoids using group theory since I haven't even learned about it yet and I understand it even worse than I understand ring theory? Also, the fact that this question is probably so basic shows my lack of understanding of ring theory.
Best Answer
Since $R$ is unital, you know it has a $1$. Since it has character $0$, you know that $1$, $1+1$, $1+1+1$, etc. are all distinct elements.
If you prefer proofs by contradiction, if the ring were finite, then by pigeonhole we must have $\underbrace{1+1+\cdots+1}_n = \underbrace{1+1+\cdots+1}_m$ for some $n \not = m$. Without loss of generality, $n > m$, and so $\underbrace{1+1+\cdots+1}_{n-m} = 0$, and $R$ has character $n-m$.
Hope this helps ^_^