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Q:

starter will not stop running on 1998 Dodge Neon

Rookie cbe0621eac06868b3efe0d8d1d3611e23c60d3114864ea2ec19a68cfbd3eebab
new starter will notstop battery ligtis on
(1) Answer
If the starter keeps running without the small START signal wire connected, the starter's faulty. If it stops running when you disconnect the START wire, then the electrical system in the car is faulty. With your meter or testlight, check that the small START wire is energized only with the key in the START position.
Then check why the charging system light is on. It's supposed to indicate when the charging system's not working.