Our DNN website (version 05.01.02 and 05.01.01) shows the Edit icon (with Pencil) to restricted users (e.g. non-admin/authors). The security permissions for the users doesn’t allow edit / update text of the HTML module control. 

Clicking the edit icon shows the edit tool strip, but dosen't can’t save any changes as the user doesn’t have permissions to do so. The real issue here is the Edit icon is displayed and I want to disable it completely.

Thanks.

Thank you Sebastian for your answer - I had already checked the security settings at the page and module level and have explicitly denied ‘edit’ access for the logged in user. But the Edit icon still appears when it shouldn't. The interesting thing is that when the user clicks the edit icon they can't save any changes as they don't have permission to do so.

I've been wrestling with this forever and I think I finally solved it.  It appears to be a module caching issue.  The default is like 3600 or something, but change it to 0.  To do this, edit the announcements module > Page Settings > Basic Settings and change the Cache Time field to 0.