Administrator accounts have 2 permission levels.
1. User, which is used to run most processes in for security reason.
2. Administrator, which will prompt you with a confirmation dialog whenever you try and do something which needs this permission.
In fact, if you try and do something that needs 2 as a normal user it will prompt you with a login box for an administrator account. The only difference between a normal user and administrator is that you do not need to log in as an administrator to grant administrator permission (as you are already logged in).
It is much more likely your problem with flash player is caused by incorrectly set up security programs than because your administrator account is not an administrator.