Defined: To rankle is to cause someone to have long-lasting annoyance, anger or irritation. When you say something to someone that you know will cause upset and anger, this is an example of a time when you rankle someone. VERB (Your Dictionary.com)
To cause persistent irritation or resentment.
To become sore or inflamed; fester.
(The American Heritage® Dictionary of the English Language, 4th edition Copyright © 2010 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.)
“It rankles some people that he got the promotion, knowing he’s not at all qualified.”
Interestingly enough, this word comes from the Latin word for “a little snake.” Read more here.