While driving after the motor gets hot it will cut off as if you had cut it off but it will crank back up and go a little father then cut off again

have alt checked,when its hot the alt regulator can malfunction and when volts get lower than 11.something the car will shut off.when you restart it the regulator resets till it gets hot again. i pulled my hair out with this problem on my tahoe

Sounds like a "HOT SOAK" problem.
Starter gets hot,it lacks power to crank the engine.

Possible "Vapor lock".

Check for engine codes.Possible bad temp sensor for ECU.

If fuel delivery is suspect ,try starting fluid into intake when "no start" occurs to diagnose fuel problem.

it can be a bad coolant temp sensor or the a fuel pump. I see a lot of fuel pumps get hot and shut off. are there any codes?

Normally the problem is electrical, something gets hot and looses contact. Could be a relay, coil, sensor, or several other components.
When it will not start you will need to figure out if you are missing spark to the plugs or fuel to the injectors to proceed.

Question: Does this problem occur after car is hot?
and if you let it cool will it restart and run till it gets hot again then shut off.

Sounds like you need a fuel pump. There is an access panel in the floor of the car to get to the pump.

It may be the CRANK SENSOR ,i had a car one time,it started fine,and drove fine one day i drove it about15 miles,stopped ,got back in it, started it, drove about 5 miles ,and cuts of with a little hesitation,it cranks but does not run, i sit there for about 20 minutes,crank it, it starts back ,i drive it back another 15 miles ,and runs great ** the crank sensor gives the signal to the ECM if the ECM ,never get that signal ,it does n't know to crank the car,or what cylinder to start with,check with your saturn ealer ,and get a second opinon

Replace the TFI module. Ford back in 1986 thur 1993 had problems like the one you wrote about. TFI module gets hot and shuts car or truck off then when it cools sometmies starts and sometimes not.

http://www.autozone.com/shopping/repairGuide.htm?pageId=0900c15280037c51


Most likly 2.4 SOHC

I will never buy a mazda again. They have bad transmissions and engines. What is worse they use the same parts as Ford. How in the world do you combine a japanese and american parts into one engine? It keeps breaking down, thats what happens. I would ride it out since you are already gettin an engine but if you haven't purchased it, sell it and put that money towards another car, the dealer has used you like a gold digger.