Q: My mother likes to fight and she always has to be "right." How can I deal with the argumentative behavior?
A: The best response to cantankerousness is no response at all. Most people who like to argue do so because it tends to evoke a strong emotional reaction from others—they do it to get a rise out of you.
Don't take the bait. If you respond to a challenge minimally, with vague, brief replies and a neutral emotional tone, your mother will probably drop the subject pretty quickly. Show focused interest in response to non-controversial subjects or those on which you really do agree, and you'll have more positive interactions. Bottom line: To increase desirable behavior, reward it.
