I read a Hadith where the Prophet, sallallaahu alayhi wa sallam, was said to have been riding his camel when he passed by one of his companions' wife. He then offered to give her a ride but she refused when she remembered her husband's jealousy. Is this Hadeeth authentic? And if it is, does it mean that we are allowed to offer a ride, in a car or motorcycle, to a woman we know, even if she is not a Mahram?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

The narration of the Prophet that the questioner referred to in the question is authentic, and is reported by Al-Bukhari and Muslim and narrated by Asmaa' Bint Abu Bakr who said: "When Az-Zubayr married me, he had no real property or any slave or anything else except a horse. I used to feed and train his horse. One day, while I was coming with the date stones on my head, I met Allaah's Apostle along with some companions. He called me and then, (directing his camel to kneel down) said, "Ikh! Ikh!" so as to make me ride behind him (on his camel). I felt shy to travel with the men and remembered Az-Zubayr and his sense of jealousy, as he was one of those people who had the greatest sense of jealousy..." This narration is not an evidence of the permissibility for a woman to be given a ride by a marriageable man in a car, or a motor bike, and it does not mean so for many reasons:

Firstly, it is probable that the Prophet directed his camel to kneel down so that she will get on it and he would walk or get on a camel with one of his companions. Her saying 'so as to make me ride behind him' is what she herself understood and the Prophet did not say so.

Secondly, it is not a matter of concern if she mounted a camel with the Prophet because he is the Messenger of Allaah, and Allaah says (interpretation of meaning): {The Prophet is more worthy of the believers than themselves}[33:6].

Thirdly, Al-Qaadhi 'Iyaadh said explaining what has been mentioned: 'This is peculiar to the Prophet - i.e. giving a ride to a marriageable woman - and he is different than other people, he has ordered us to keep a distance from women and he used to stay far from women so that people would imitate him. But this incident was peculiar to him, as she was the daughter of Abu Bakr and the sister of 'Aa'ishah and the wife of Az-Zubayr so she was like one of his family and wives, and he was the most chaste of people.' In addition, some scholars said that it is permissible to give a lift to a marriageable woman in a car with pious men if she is on the street. However, the most preponderant opinion is that it is not permissible, especially in corrupt times when the faith is weak, except if there is a need and there is no other way except to give her a lift on the condition that neither of the two will be tempted. 

Allaah knows best.