As a general rule it is permissible for a man to wear a ring since the Prophet (saw) would wear a ring from silver (agreed upon). Some scholars were of the view that the Prophet (saw) would only wear the ring since he used it as a seal for his (saw) letters, thus it was not really used for zeenah (adornment). However, the strongest view according to many scholars is that it is permissible to wear even if it not used as a seal since the companions used to wear them and the Prophet (saw) did not rebuke them for that. Thus, if it is from the customs of the people to wear rings, then it is fine as long as it does not entail anything prohibited, such as using gold instead of silver.
As for your mentioning of ‘wedding ring’ then I assume that you are using the term ‘wedding’ because you possibly saw the ring being worn on the left hand.
Ibn Rajab states in his treatise on rings:
“It is permissible to wear the ring on the right and left hand. The scholars however differed on which hand it is more virtuous to wear it on.
A group of them said that it is more virtuous to wear it on the left hand. This was stated opinion of Imam Ahmad according to the narration of Salih: “Wearing the ring on the left hand is more beloved to me.” And he also said: “It is more established and stronger.” The same view has also been transmitted from al Fadl bin Ziyaad and it is also the opinion of the mathhab of Imam Malik. It is also reported that he (Malik) would wear the ring on his left hand. The same has also been reported from ash-Shaafi’ee…
It was also reported in Sahih Muslim that Anas (ra) said: “The ring of the Prophet (saw) was on this hand, and he pointed to his little finger on his left.” And it was also reported in Abu Dawud that Ibn ‘Umar stated that the Prophet (saw) used to wear his ring on his left” [considered shaadh by al-Albani: 4227]…”
It has also been authentically reported that both al Hasan and al Hussain would wear their rings on their left hands. (Tirmidhi: 1743).
It has also been mentioned by the later huffadh (of Hadeeth) that wearing the ring on the left hand was reported to be the practice of most companions and taabi’een.
However, despite all of that, many scholars held that wearing it on the right hand is more virtuous. This was the view of Ibn ‘Abbas.
Their evidences are as follows:
At-Tirmidhi & others narrate that Hammad bin Salamah said: “I saw Ibn Abi Raafi’ with a ring on his right hand and I asked him about that and he said: “I saw ‘Abdullah bin Ja’far with a ring on his right hand and he said: “The Prophet (saw) used to wear a ring on his right hand.” (1744). Imam Bukhari said: “This is the most authentic thing narrated regarding this matter.
The equivalent has also been reported from Ibn ‘Abbas & others as well. (Abu Dawud: 8/175, Ibn Maajah 3647 [saheeh according to al-Albani].
The kalaam of the scholars regarding these two variant groups of narrations is very extensive, and it is not the place to discuss it here. But in generality, we can see that both views have their evidences and thus no inkaar (rejection) should be shown to each of the two views.
Interestingly though, some Hanbali scholars state that the practice of wearing the ring on the right hand was abrogated and that wearing the ring on the left was the final state reported from the Prophet (saw). This is backed by the narration of Ibn ‘Umar: “The Prophet (saw) used to wear a gold ring on his right hand and then the people started to do the same and then he threw it off and said: I will never wear it again” and thus the people did the same.” (Reported by al Tirmidhi in ash-Shamaa’il.) also, according to the narration of Sulayman bin Muhammad al Faaqilaani it mentions that he (saw) then started wearing his ring on his left hand.” (see ahkaam al khawaateem: p.92).
It is also permissible to wear the ring on the 2nd to last finger or thumb and more preferable to wear it on the last finger. As for the middle or index finger, then it is disliked by most scholars and some stated that it is prohibited.
And Allah knows best.