zsh renaming sets of files

I want to rename a couple of dozen files in a folder that contains over 100 files. The files to be renamed all have existing names that contain a single pattern. For example,

abcSOURCEFILE1

xyzSOURCEFILE2

jklSOURCEFILE3

etc. should be renamed to

SOURCEFILE1

SOURCEFILE2

SOURCEFILE3

etc.

Is there a single command that can do this? Something like "mv *SOURCEFILE? SOURCEFILE? Is the actual "rename" command useful? Or do I need a script?

TIA,

Mark

P.S. Is there a tutorial somewhere that shows how to do this? When I searched the Internet, I found either examples that were too simple, or descriptions that were too complex (because they were trying to cover the full power of globbing, or they distributed the infomation needed to construct a solution over several documents - the "man" pages have this problem, or both).