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Yes, legally and for the computer to function correctly you will need a catalytic converter.
That is the sensor after the catalytic converter.
With gas engine, yes. Not sure about diesels. Diesel, yes
Yes
On a gas engine there is one on each side of the catalytic converter. The diesel doesn't have any.
Yes, your 2003 ram diesel has a catalytic converter. The converter is located under the cab.
That is the downstream O2 senosr and it located after the catalytic converter on the passenger side
Muffler should be after the cat.
If someone has not removed it, Then yes it is before the muffler.
In 2001 the Ram 2500 diesel did not come with a cat. it has a resonator and a muffler. the thing under the cab that looks like a cat is actually a resonator. so no they are not required
1996 and later gasoline engines are equipped with pre and post catalyst O2 sensors. The sensors are located in the exhaust mainfold one before the catalytic converter and one after.
If you remove a catalytic converter, you are breaking the law. It is a $2500 fine. It will not run right anyway, because of the oxygen sensor after the catalytic system. You are just going to get horrible fuel mileage if the cat is removed, and a check engine light that is annoying as hell.