# Notes - Linear Algebra MT23, Integral domains

### Flashcards

What is an integral domain?

A non-zero ring s.t. if $a, b \in R$ and $ab = 0$, then either $a = 0$ or $b = 0$.

Quickly prove that all fields are integral domains.

\[\begin{aligned}
&ab = 0 \quad\text{wlog assume } a \ne 0\\\\
\implies& a^{-1}ab = a^{-1}0 \\\\
\implies& b = 0
\end{aligned}\]