Notes - Rings and Modules HT24, Integral domains

Created: January 18, 2024

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 $a$.



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