What is the ruling on shaking hands with a non-mahram woman, and does it invalidate the ablution?

Q: The brother: M. R. Y - from Daraa, Syria, says in his question: I was in one of the gatherings, and there was a shaykh among us. After a while, a woman entered and shook hands with everyone. After the mu'adhdhin called for the 'Isha' prayer, the shaykh stood up and prayed. After the prayer, I asked him about the permissibility of a man praying after a woman had shaken his hand without him performing wudu'. He answered me: If there is no bad intention from either of them, then it is permissible for either of them to pray without wudu'. What is your opinion of his statement?

A: It is not permissible for a man to shake hands with a non-mahram woman; because of his saying (peace and blessings be upon him): and the saying of 'A'ishah (may Allah be pleased with her): and because of the causes of fitnah in shaking hands with them.
As for touching a woman, there is scholarly disagreement regarding whether it invalidates the wudu', and the correct view is that it does not invalidate the wudu'; whether it is done with desire or without desire. The Messenger (peace and blessings be upon him) kissed some of his wives and then prayed without performing wudu'. Also, the original ruling is that wudu' is not invalidated except by clear, authentic evidence, and there is no clear, authentic evidence in this matter indicating that touching a woman invalidates the wudu'. Furthermore, this is a matter that affects every household, so if touching a woman invalidated the wudu', the Messenger (peace and blessings be upon him) would have explained it clearly.
As for His saying (Glorified and Exalted is He): [al-Ma'idah: 6], the intended meaning is intercourse, as stated by Ibn 'Abbas and a group of scholars, and the intended meaning is not touching with the hand, according to the most correct of the scholars' opinions.