Good morning, I am using ArcGIS 10.1, previously used version 9.1. When I add the base map (Imagery) the coordinate system of the data in use is changed. As can keep the coordinate system and that the Base Map (Imagery) will fit?
If I am understanding you correctly... When you add a data layer to a new blank map document, the data frame will adopt the coordinate system of the first data layer. For the online base maps, this is web Mercator. Then, if you want to be in a different coordinate system, simply right click on the data frame, select the coordinate system tab and select a new coord sys.
All data added to the frame will be reprojected on-the-fly into whatever system the data frame is using. The rider to this is when the input data has "no projection" or has an incorrectly specified system. But that's another debate. Cheers, Neil