Below are a few propositions, most of which most likely only require short, fairly easy proofs.
- Prove that every unital ring of characteristic zero is infinite.
- Is an example of an infinite ring of characteristic 10 the infinite Cartesian product of the set of integers modulo 10?
- Prove that every integral domain has characteristic zero or p, where p is prime.
Here's what I've come up with so far (I apologize if some of it is unclear due to my lack of understanding):
1) Let $R$ be a unital ring of characteristic zero. Assume $R$ is finite. By assumption, we can write $R$ as $R = (a_{1}, a_{2}, ...,a_{n}),$ where $n\in\mathbb{N}$ is finite. Since $R$ is unital, we may assume without loss of generality that $a_{1} = 1$. Since $R$ is of characteristic $0$, then $\not\exists n\in \mathbb{N} \space(n\cdot 1 = 0)$. $1\neq 0$ in $R$. I'm stuck here. I know I should show that there exists another element in $R$, thus contradicting the finiteness of $R$ and proving that it is infinite. But I can't seem to find a way to do that.
1) $R$ is a group under addition. Recall that if $G$ is a group and $m$ is the order of $G$, then $g^m$ is the identity for all $g \in G$. Alternatively, since $R$ is finite there exist distinct $m,n \in \mathbb{N}$ such that $m \cdot 1 = n \cdot 1$ (or else $\{m \cdot 1 : m \in \mathbb{N}\}$ is infinite). Then $(m - n) \cdot 1 = 0$.
2) Yes.
3) Suppose $I$ is an integral domain of composite characteristic $n$. Then $n = ab$ for positive integers $a,b$ with $a,b < n$. For $m \in \mathbb{N}$ let $m_I$ denote the element of $I$ obtained by adding the multiplicative identity to itself $m$ times. Show that $a_I,b_I \neq 0$ and $a_I \cdot b_I = 0$. The key here is to check that $(ab)_I = a_I \cdot b_I$, where $ab$ denotes the product of the integers $a$ and $b$ and $\cdot$ is multiplication in $I$.