Hadith: I went to the Prophet (may Allah's peace and blessings be upon him) with my father. The Messenger of Allah (may Allah's peace and blessings be upon him) then asked my father: Is this your son? He replied: Yes, by the Lord of the Ka‘bah

عن أبي رِمْثَةَ رضي الله عنه قال: انطلقت مع أبي نحو النبي صلى الله عليه وسلم ثم إن رسول الله صلى الله عليه وسلم ، قال لأبي: «ابنك هذا؟» قال: إِي ورَبِّ الكعبة، قال: «حقا؟» قال: أشهد به، قال: فتبسم رسول الله صلى الله عليه وسلم ضاحكا من ثَبْتِ شَبَهِي في أبي، ومِنْ حَلِفِ أَبِي عَلَيَّ، ثم قال: «أما إنه لا يَجْني عليك، ولا تَجْني عليه»، وقرأ رسول الله صلى الله عليه وسلم : {ولا تَزِرُ وَازِرَةٌ وِزْرَ أخرى} [الأنعام: 164].
[صحيح] - [رواه أبو داود والنسائي وأحمد والدارمي]

Abu Rimthah reported: I went to the Prophet (may Allah's peace and blessings be upon him) with my father. The Messenger of Allah (may Allah's peace and blessings be upon him) then asked my father: "Is this your son?" He replied: "Yes, by the Lord of the Ka‘bah." He again said: "Is it true?" He said: "I bear witness to it." The Messenger of Allah (may Allah's peace and blessings be upon him) then smiled due to my resemblance with my father and because my father took an oath that I was his son. He then said: "Indeed, He will not bear your faults (sins), and you will not bear his." The Messenger of Allah (may Allah's peace and blessings be upon him) recited the verse: {And no bearer of burdens will bear the burden of another.} [Sūrat al-An‘ām: 164]
Authentic hadith - [An-Nasaa’i]

Explanation

Abu Rimthah (may Allah be pleased with him) reports that he went with his father to the Prophet (may Allah's peace and blessings be upon him). So the Messenger of Allah (may Allah's peace and blessings be upon him) asked the father if Abu Rimthah was his son. The father affirmed that it was so and swore on it. Therefore, the Messenger of Allah (may Allah's peace and blessings be upon him) smiled at the father's response. He told him that none, neither the relative, nor the distant one, is to be held accountable for someone else's crime, even fathers and children. The criminal is to be held accountable for his own crime only. In this respect, Allah, the Almighty says: {And no bearer of burdens will bear the burden of another} [Sūrat al-An‘ām: 164]. It was common in pre-Islamic times, though, that one would be held accountable for the crime of his relative. However, in Islam such a practice was revoked. A possible objection here is that Islamic law enjoins upon the male paternal relatives of a Muslim to pay the blood money in case their relative kills someone unintentionally or wounds him! The answer is that this case belongs to mutual support among the Muslims rather than being held accountable for the crimes of others. Also, the relatives inherit the perpetrator when he dies, so they should pay for the blood money on his behalf if he makes a fault.