Wa `alaykum as-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
1- Wearing silver rings is permissible for Muslim men as well as women.
2- However, there is no authentic evidence that doing that has a special merit during the month Rajab.
In his response to the question, Sheikh Muhammad Saleh Al-Munajjid, the prominent Saudi Islamic lecturer and author, stated,
It is permissible for both men and women to wear silver rings. Al-Bukhari and Muslim narrated that Anas ibn Malik said, “When the Prophet (peace be upon him) sent a letter or wanted to send a letter (to the non-Arab rulers), it was said to him, ‘They will not read any letter unless it has a seal on it’. So he had a silver ring made and engraved on it Muhammad Rasul Allah (Muhammad Allah’s Messenger). It is as if I can see its brightness on his hand.”
An-Nawawi (may Allah have mercy on him) said in al-Majmu`:
“It is permissible for a woman, married or not, to wear a ring of silver as it is permissible for her to wear a ring of gold. There is consensus on this point. There is nothing disliked (makruh) about that, and there is no difference of scholarly opinion concerning this matter. Imam Al-Khattabi said, “It is makruh for her to wear a ring of silver, because that is one of the symbols of men. So if she cannot find a ring of gold, let her color it yellow with saffron and shabaha.”
But what he said is wrong and there is no basis for it; the correct view is that it is not makruh. Then he said, “It is permissible for a man to wear a ring of silver, whether he holds a position of authority or not. There is a scholarly consensus on this point. With regard to what has been narrated from some of the earlier scholars of Ash-Sham, that it is makruh for anyone other than a ruler to wear it, this is an odd view and it is be rejected on the basis of the texts and the consensus of the earlier generations. Al-Anbari and others narrated that there was a scholarly consensus on this matter.”
It is also permissible to engrave or write things on a ring, but there is no basis for doing that specifically in the month of Rajab, or for wearing the ring in the month of Rajab believing that it will bring one close to Allah, or that there is a particular virtue in wearing it during this month. This is an innovation and is wrong.
You should be aware of writing anything on the ring that it is claimed will bring good luck or ward off the evil eye, envy and the jinn, and so on.
To sum up, in principle, there is nothing wrong with wearing rings and engraving things of them. What is forbidden is seeking to draw closer to Allah by doing that, or singling out a specific time for wearing it, or seeking barakah (blessing) from the ring, or using it as an amulet.
Allah Almighty knows best.