Fatimah is the daughter of the leader of mankind, Prophet Muhammad (peace and blessings be upon him), and mother of Al-Hasan and Al-Husayn. She was born shortly before the beginning of the mission of the Prophet and married `Ali ibn Abu Talib (may Allah be pleased with him) after the Battle of Badr. The Prophet (peace and blessings be upon him) loved her so much and honored her. She was patient, devoted to Islam, generous, chaste, devout, and thankful to Allah. `A’ishah is the daughter of Abu Bakr (may Allah be pleased with him) and the wife of the Prophet (peace and blessings be upon him). The Prophet (peace be upon him) loved her more than any one of his wives except Khadijah (may Allah be pleased with her), and he (peace be upon him) stated that she is perfect in her understanding. As for `Ali, he (may Allah be pleased with him) was brought up by the Prophet and learned from the Prophet the good manners.

It is unbelievable that `A’ishah (may Allah be pleased with her), the mother of the believers and wife of the Prophet, had a bad relation with the Prophet’s daughter or his cousin `Ali. A Muslim should pay no attention to the false reports invented by those who hold grudge against the Companions of the Prophet and try to substantiate their lies with fabricated reports, which they attribute falsely to the ever purest generation throughout the history of humanity.

the Companions of the Prophet, who sacrificed themselves for the sake of Islam and migrated from their homelands, were not to dispute for the transient matters of the world. We should think highly of them because they are our reverend predecessors, and admitting falsehood about them is tantamount to admitting falsehood to Islam because they are the ones who transmitted it to us .

Sheikh `Abdulla ibn Sardar, lecturer at the Umawi Mosque in Madinah, stated,

In fact, `A’ishah had a good and unblemished relationship with Fatimah (may Allah be pleased with them both). `A’ishah would often praise Fatimah’s good character and relate hadiths about how good she was. `A’ishah said, “I have never known anyone who was more like Allah’s Messenger (peace and blessings be upon him) in his disposition and understanding than Fatimah.”
also, the relationship between `Ali (may Allah be pleased with him) and `A’ishah was generally good as well, except for something that happened during the Battle of the Camel. The cause of this conflict was that some of the Companions, including `A’ishah, were demanding that the retribution for killing `Uthman ibn `Affan (may Allah be pleased with him) be exacted from his murderers. `Ali and those with him wanted to postpone carrying out the retribution due to the difficult political circumstances that `Ali had to deal with at that time. They all went to Basrah and agreed not to fight.

however, a group of troublemakers attempted to start a war. So, they acted on their own and attacked the companions of `A’ishah in the next morning. The Companions in both sides managed to stop the fighting and bring things back under control on the same day. `Ali then praised `A’ishah, and she praised him. May Allah be pleased with them both!
I would like to point out something very important. Allah is not going to ask us on the Day of Judgment about the Companions’ relationships with one other. That is why the scholars of Ahl As-Sunnah believe that Muslims should not occupy themselves with these matters. Such preoccupations do nothing to help a Muslim in his or her beliefs or in his or her worship.