A ticket mark "Resolved" will stay in your ticket view until you or the client "Close" the ticket.
Note: You can turn this off my navigating to Setup -> Tickets -> Status Types -> Choosing "Resolved" Status Type -> Uncheck the "Include in My/Group Tickets Filter" option. You can also choose to set the amount of time to "Automatically Close Ticket After" option to Auto Close after a said amount of time.
You can also apply a filter to your view to show just "Opened" tickets - thus purging your view of the "Resolved" (and all other status's) tickets.
I hope this helps.
The "Automatically close ticket After X days" is the best option I think. It nice because the user still gets the this item is resoved is it fixed email but will permantly close after your set amount of business days what we initlaly found was many users did not understand or realize they had to click yes to mark the issue as fixed and closed. Hiding them just leaves them open for reproting pourposes.