...because in your inner loop $TEMP and $TEMP1 have the same value.

You may want to try it like this:

#!/usr/bin/perl -w

use strict;

my $dir=".";
opendir CONCEPT, $dir || die "no such dir";
while (my $temp = readdir(CONCEPT)){
  if ($temp =~ /^(A-\d+-\d+)/) { 
    rename $temp, $1 or die "error $! renaming $temp to $1";
  }
}
closedir CONCEPT;
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