Even if my car has power door locks, the guys at the Mazda dealer say they can not do it.They've tried it on another cas same as mine and it did not work.

Strange!
My 2007 Mazda3 had remote keyless entry as standard built into a key (folding key, looks something like this http://caradvice.com.au/wp-content/uploads/2007/10/tc-i30-fob.thumbnail.jpg).
Now, my car was made for European market but it was made in the same Japan factory as all Mazda3s so l find it hard to believe that the locking systems would different. It is just some features may have been disabled.

l had a rather sophisticated alarm & shock sensors fitted, it was all paired with factory engine immobilizer. Now l can lock & unlock my car either with a Mazda key or by a separate remote dong. It did not affect my waranty, in fact, it was fitted by an independent company that is ''endorsed'' by my dealer.

l had the same alarm fitted to my old 1998 Mazda 626 that did not have remote keyless entry as standard & again it all worked. Same with my father is 2002 Mazda6 & his new 2007 Mazdaspeed6.

This is what l had fitted
http://jablotron.cz/en/car.php?pid=products/ca-320
It is remotely controlled alarm/keyless entry system.
Some of Jablotron is products r also available in the USA, but l am sure there must be similar products on the US market.

Try an alarm specialist or similar company. If it works on my car it MUST work on yours too.

no major production car has a factory alarm
just think about this... for EXAMPLE mazda makes around 1Million cars, each alarm cost 15$ they will spend 15Millions in alarms? they r not

the insurance companies demand each car to be sold has alarm & dealers have to install them, each dealer install alarms to their cars, so if ur dealer says they cant is because they r using an alarm non compatible to keyless mode

in other words u have to install a new alarm to ur car