Once the join is performed in ArcMap, right click the joined layer and go down to Data -> Export
In arcmap, when a join is performed, it is only in-memory so-to-speak. The join will only exist inside the map document it was performed on.
Exporting using the step above will make the join permanent with the joined fields now part of the attribute table. Note that this does create a new feature class/shapefile (or duplicate depending on how your workflow is currently being defined) so perhaps the old layer could be considered residual and be safe for the trashbin?
In earlier versions of arc, I remember seeing that these temp fields were denoted with an Asterix * attached to the columns header. I think now, they are expressed more in a SQL/pythonic manner, example: layername.FieldName
That formatting of the attribute header will disappear when the join has become permanent.
