I want to build a very simple application with Arduino and some Ethernet adapter. I was thinking of the Ethernet shield but its too costly and anyway I don't plan to use but a small portion of its functionality. I just want a device that connects to the Internet (being connected directly on a router, without the need for a computer) so that I can control it over the Internet and turn on/off a relay switch. Is there a way to accomplish that with Arduino and without the Ethernet shield?
Pretty much the only functionality the Ethernet Shield has is Internet access and that seems to be what you want. Does your router have any connection option other than Ethernet? I suppose you could use WiFi but I'd be surprised it that was any cheaper.
Have you looked at the Arduino Ethernet? It includes the Ethernet connector without a shield. It costs about $35 more than the UNO.
I was thinking of the Ethernet shield but its too costly
I don't think you can do what you want to do for less than the below boards.