You don't have to use Flex builder it just makes taking your Flex app to the desktop easier buy using the integrated tools and SDK.
You can also write AIR apps using other Adobe technologies Flash CS3 or Dreamweaver CS3 (XHTML & JavaScript).
If you don't want to purchase any IDE to build your AIR apps you can just get the
AIR SDK and use any text editor to do your code.
I think there are plugins for IDE's like APTANA and FlashDevelop. There might be others too.