What is the ruling on a man wearing a silver wedding ring with the intention of reducing annoyance at work and in public, and is it disliked to wear it on the right hand in order to differ from others?

There is no harm for the questioner in wearing a silver ring, as he is not wearing it for an engagement. The Sunnah permits men to wear silver rings. It is narrated in Sahih Muslim that the Messenger of Allah, peace and blessings be upon him, wore a silver ring.