Avhienda has the ability to sense Ter'angreal, but what about Sa'angreal meant for men? How possible could it be that Lan's heavy golden ring is really the Ring of Tamerlyn passed down in his family from generation to generation? I know nothing's been said that could support this theory but it's the only ring in the story that has been there from the beginning and it just seems like the perfect candidate

If I remember right, Lan's ring has been melted down and remade multiple times over the generations so it's doubtful that it's the Ring of Tamyrlin. Or at least, that it's still a sa'angreal if it was (and if the RoT even was).