Yes. Your employer would be well within its rights to prevent you using its computer equipment and connectivity for personal use, be it personal email, online shopping, or using social networking sites.
Some employers will be happy for you to make personal use of the internet during clearly defined periods, such as breaks, or after hours. Others who want to structure work in more flexible ways may be happy for you to take occasional breaks when you wish, so long as you make up the time overall. It depends on the management style of your employer.
This is also relevant to use of a work email address to register for services. You need to know whether you're permitted to use the email address in this way. Unless you are using Facebook officially as part of your job, it would be more appropriate to use a private e-mail address.
24 Feb 2012: When do you start getting paid?
The Interns: Getting treated like a dogsbody?