The Morgenstern family ring is the family ring of the Morgensterns. The ring is made out of silver. It has an "M" on it to stand for the family name, and since the family's symbol is a falling star, most of the rings also have a pattern of stars around the band.
In 2007, Jace Herondale, then known as Jace Wayland, wore the ring given to him by the man who raised him. He wore his ring, a plain band with only the family's initial and no falling star symbol, upside down, believing the M on the ring was actually a W that stood for Wayland. The ring can be used to summon or call for Valentine's presence when twisted three times, which Hodge Starkweather used to do so. Presumably Valentine's other men, including Malachi Dieudonne, also had access to similar rings for this very purpose.
After Jace falsely realized that he was Valentine's son, Jace chose to keep the ring but wore it on a chain, unwilling to neither wear it nor lose it. However, he did eventually give the ring to Clary, Valentine's true daughter, wanting it to remain with someone he loved and trusted. Clary kept the ring and has since worn it on a chain around her neck, like Jace did.
Once, Kaelie Whitewillow inquired about why she wore the ring around her neck, wondering if it was a tribute to her father. Clary denied this and made it clear that she was wearing it for Jace. When Jace went missing, Clary lent the ring to Magnus Bane to help him in his attempts to track Jace, to no avail. When Jace was rescued some time later, he had Alec get the ring from Magnus. Jace then returned it to Clary, who agreed to continue wearing it.
Aside from Clary, her brother Jonathan also had his own Morgenstern ring. While Sebastian usually wore the ring on his finger, it was on a chain on his neck when he was burned in Alicante. The ring remained intact even as Jonathan was reduced to ashes. Afterwards, Clary threw the ring along with Jonathan's ashes into the Lake Lyn in Idris.