Praise be to Allah.
It is permissible for a man to wear a silver ring, as it is also permissible for a woman to do so.
Al-Bukhaari (65) and Muslim (2092) narrated that Anas ibn Maalik (may Allah be pleased with him) said: When the Prophet (blessings and peace of Allah 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 ring of silver made, and engraved on it Muhammad Rasool Allah (Muhammad the Messenger of Allah). It is as if I can see its brightness on his hand.
An-Nawawi (may Allah have mercy on him) aid in al-Majmoo ‘ (4/340):
it is permissible for a woman, married or otherwise, 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 makrooh about that, and there is no difference of scholarly opinion concerning this matter. Al-Khattaabi said: It is makrooh for her to wear a ring of silver, because that is one of the symbols [?] of men. So if she cannot find a friend of gold, let her colour 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 makrooh.
Then he said: It is permissible for a man to wear a ring of silver, whether he holds a position of authority or otherwise. There is scholarly consensus on this point. With regard to what has been narrated from some of the earlier scholars of ash-Shaam, that it is makrooh for anyone other than a ruler to wear it, this is an odd view and is to be rejected on the basis of the texts and the consensus of the earlier generations. Al-Anbari and others narrated that there was scholarly consensus on this matter. End quote.
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 months of Rajab in the belief 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.
And Allah knows best.