How to separate in object mode? I have successfully separated two rings out of a UV sphere using "separate" (Y key) option in Edit mode. But as soon as I switch to Object mode, there is no separation and the objected is treated as one. What can I do to separate it in Object mode for real?
You have made two different meshes but they are still parts of the same object. To make a new object, go back in Edit mode, select the separate mesh with L, then press P > Selection, this part will be a new object.

By the way, to create a separate object you don't have to separate the part of the mesh with Y or V, you can just select the part of the mesh you want to separate, then P (separate) > Selection.