Is it permissible for him to engrave the name of Allaah on his ring?

There is nothing wrong with a man wearing a ring of silver, and there is nothing wrong with writing the name of Allaah and the like on it. 

Al-Bukhaari (5877) and Muslim (2092) narrated from Anas ibn Maalik (may Allaah be pleased with him) that the Messenger of Allaah (peace and blessings of Allaah be upon him) acquired a ring of silver, on which was engraved Muhammad Rasool Allaah, and he said to the people: “I have acquired a ring of silver and engraved on it Muhammad Rasool Allaah, and no one should have an engraving like this.” 

This indicates that it is permissible to engrave the name of Allaah on a ring. What the Prophet (peace and blessings of Allaah be upon him) forbade was for anyone else to engrave the same as he had, because it contained his name and description. Moreover, he acquired this ring in order  to use it as a seal, so it was a sign that was unique to him and distinguished him from others. If it were permissible for anyone to have the same engraving, that purpose would be defeated. This was stated by al-Haafiz in al-Fath. 

It was narrated that many of the salaf had phrases engraved on their rings that included the name of Allaah, some of which were mentioned by al-Haafiz in al-Fath, who said: 

In al-Musannaf, Ibn Abi Shaybah narrated that Ibn ‘Umar had engraved on his ring “ ‘Abd-Allaah ibn ‘Umar.” Ibn Abi Shaybah narrated that Hudhayfah and Abu ‘Ubaydah each had a ring on which was engraved “al-hamdu Lillaah (praise be to Allaah).” ‘Ali had a ring on which it said “Allaah al-Malik (Allaah the Sovereign)”. Ibraaheem al-Nakha’i had a ring on which it said “Billaah (by the help of Allaah).” Masrooq had a ring on which it said “Bismillaah (in the name of Allaah).” Abu Ja’far al-Baaqir had a ring on which it said “al-‘Izzah Lillaah (Glory belongs to Allaah).” It was narrated from al-Hasan and al-Husayn that there is nothing wrong with engraving the name of Allaah on a ring. Al-Nawawi said: this is the view of the majority. It was narrated from Ibn Sireen and some scholars that it is makrooh. End quote. 

Ibn Abi Shaybah narrated with a saheeh isnaad from Ibn Sireen that he did not see anything wrong with a man writing on his ring “Hasbi Allaah (Allaah is sufficient for me)” and the like. This indicates that it is not proven that he regarded it as makrooh. We may reconcile the reports by saying that it is regarded as makrooh when there is the fear that it may be worn by one who is junub or menstruating, or that a person may clean himself after reliving himself using the hand on which it is worn, and it is permissible when there is no risk of that, so it is only regarded as makrooh because of the danger of it being exposed to disrespect. End quote. 

And Allaah knows best.