A shut up ring is an engagement ring given to a partner to get them to shut up or stop pushing for marriage.
Usually the one who gives the ring is the one who is reluctant to get married/engaged, so they provide a ring to the one demanding so they will stop talking about it.
Person 1: "Lisa wanted Jason to propose so badly, but he didn't want marrige yet."
Person 2: "Yeah. I heard he just gave her a shut up ring so they could both be happy."