Rings and Modules HT24, Integral domains
Flashcards
Suppose $R$ is a ring. What is a zero-divisor?
$a \in R\setminus\\{0\\}$ such that $\exists b \in R$ where $ab = 0$.
Quickly prove that the charactersitic of any integral domain $R$ is either $0$ or prime.
Suppose $\text{char}(R) = n = ab$. Then $a _ R b _ R = 0$, but since $R$ is an integral domain, either $a _ R$ or $b _ R$ is zero, which contradicts the minimality of $n$.
What construction allows you to generalise the idea of the rationals to any integral domain $R$?
Consider $R \times (R \setminus \\{0\\}) / \sim$ where $(a, b) \sim (c, d) \iff ad = bc$.
Suppose:
- $R$ is an integral domain
- $R$ is finite
Quickly prove that $R$ is a field.
Fix some $a \in R \setminus \\{0\\}$, we wish to show that $a$ has a multiplicative inverse. Consider the map $\phi : R \to R$ given by $\phi(x) = ax$.
Then $\phi$ is injective, since
\[ax = ax' \implies a(x - x') = 0 \implies x - x' = 0 \implies x = x'\]Then because $\phi$ is an injective map between finite sets of the same size, it must also be surjective. Hence $\exists x$ where $ax = 1$, i.e. $x$ is the multiplicative inverse of $a$.