Internet Connection Goes To Sleep When Not In Use?

My mother's ISP is an AT&T DSL connection. She just had her PC connected directly to her modem for several years, so I eventually got her a router so when her kids and grand kids came to visit they could use wifi. It was a very basic and inexpensive Belkin model, but I don't recall the model number. The problem was the Internet connection would become inactive every time the desktop PC that was connected to router via Ethernet would go to sleep. The only way to get the Internet connection back was to wake up the desk top PC. I checked all of the router settings, googled to see if I could find similar problems, etc... and never could fix it; so I eventually set up my trusty old wrt54g at her house. It worked fine in my home for years; and when I set it up at her house, it still had all of my original network settings in it. I just changed the network name and password.But the same issue is still there. Every time the desktop PC goes to sleep... The router loses Internet connection and does not get it back until the desk top is made active again