I'm trying to get email working on my home Linux box. I can send
email, and receive mail if it's addressed with the hostname, but not
if it's addressed with only the domain name.

me@mymachine.mydomain.net <-- works
me@mydomain.net           <-- doesn't work

I want the second version to work. The error message is below. I've
disguised the names for some reason.

   ----- The following addresses had permanent fatal errors -----
<me@mydomain.net>
    (reason: 550 5.7.1 <me@mydomain.net>... Relaying denied)

   ----- Transcript of session follows -----
... while talking to mymachine.mydomain.net.:

>>>>>> RCPT To:<me@mydomain.net>

<<< 550 5.7.1 <me@mydomain.net>... Relaying denied
550 5.1.1 <me@mydomain.net>... User unknown

Outgoing mail appears to be from me@mydomain.net, as I expect/desire.

Info: Sendmail v8.12.10. This machine is the primary dns server (bind
9.2.2-P3) for mydomain.net. It's running Fedora Core 1 (RedHat's poor
relation?) It's my machine and my domain, so I can change anything I
need to.

It seems like I'm missing something simple, if there is anything
simple about sendmail, so I've priced the question accordingly. If
this requires a lot of work and consideration of my sendmail.cf, I can
go somewhat higher, or would accept a pointer to the right part of the
O'Reilly Sendmail book at the given price. Or maybe I've missed the
boat entirely and this is a DNS issue? I have one MX record:

mydomain.net. IN MX 10 mymachine.mydomain.net.

I have pretty much the same setup with RedHat9.  

In /etc/mail/local-host-names you should put in all the aliases for your machine.

like

mx1.mydomain1.com
mx2.mydomain1.com

mx.mydomain2.net

and
mydomain1.com
mydomain2.net

Perfect!

Thanks much