Engegement rings...

An engagement ring is a ring indicating that the person wearing it is engaged to be married, especially in Western cultures. ... The neologism "mangagement ring" is sometimes used for an engagement ring worn by men. In some cultures, men and women wear matching rings, and engagement rings may also be used as wedding rings.

...Once you're married, tradition dictates that your engagement band be moved back to the third finger on your left hand. When you do so, your wedding ring should remain closest to your heart (where your spouse placed it on your wedding day) and your engagement ring is placed next to the wedding ring.