What is the Islamic legal directive for children and their mother, and what is their duty towards their parents, in the event that the father irrevocably divorces their mother and then marries another woman, and the mother requests that her children not greet their father's wife, becoming angry with the children when they do so?

Righteousness towards parents and obedience to them in what is good and kind treatment of them are obligatory. However, obedience to the mother is not obligatory if it involves severing ties with the stepmother or refraining from greeting her. It is appropriate to maintain ties with the stepmother as an act of righteousness towards the father, but one can refrain from informing the mother about it to avoid displeasing her, by employing wisdom and tact.