At the risk of coming off as "preachy", the reality is (as Sandy mentioned) it's only a symbol. If it has sentimental (human) value to you, decide how much value and go from there. If you just want it whole again to toss it into a drawer, superglue away. If you feel it's worth more to you than that, look into a jewelry repair.

It is not a sin, nor any reflection on a person's faith to discard unwanted material items. Allowing yourself to believe that you're somehow being unfaithful to God by eliminating such items/stuff is on the edge of idolatry and that's a big no-no. God's a little tougher than that.