question regarding how to launch perl script on remote machine

so just to put it out there for my ideas to run more perl scripts at 
work using modules that I cannot install(whether due to lack of 
knoweldge or just don't have
the right)..

at work, we have a solaris based unix server(lets say serverK) which is 
being served as central logon for all the daily tasks.
I don't have root right and I have tried many things to install some 
modules that I wanted to run(DBI,Expect among other things) but could no 
make it work..
I also have list of linux machine behind serverK, called linux1-4 which 
btw cannot reach any of other servers out 
there(server1/server2/server3.........etc)
I DO have root right in linux machines and I am installing all the 
modules that I know I want to use..

What I am planning to do is write the scripts I want to run on SERVERK 
on linux machines and somehow just need to launch them from SERVERK.
For very simple example, if I wanted to write a perl script(using 
expect) to logon automatically to (server1-4 etc) do something.... is it 
possible to launch
a script that's stored on linux machine from SERVERK ?
I have thought of writing a simple shell script which launchs a (from 
SERVERK)ssh session into linux machines using user name with initial 
script to run a perl script and logs off..
Is that the only way?
I guess I am writing this to see if there are people with similiar 
problem....

server1           server2         server3          server4      
.....................


                                SERVERK  (this is where everyone logs on 
& need to run the program)


linux1              linux2              linux3                  linux4  
(these linux machines under my control)