Praise be to Allaah.
It is mustahabb to remove anything on which the name of Allah is written when entering the bathroom (place where one relieves oneself). If one needs to enter wearing it, then it should be covered before entering.
Ibn Qudaamah (may Allah have mercy on him) said in al-Mughni (1/109): If he wants to enter the washroom and has something with him on which the name of Allah is mentioned, it is mustahabb for him to put it down or take it off. If he keeps with him the thing that he has on which the name of Allah is mentioned, and takes care not to drop it, or turns the part of the ring on which it is written to face his palm, there is nothing wrong with that. Ahmad said: If the ring has the name of Allah on it, he should turn it to face his palm and enter the washroom. This was the view of Ishaaq and Ibn al-Musayyab, and al-Hasan and Ibn Sireen granted concessions allowing that. End quote.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: What is the ruling on entering the washroom with papers on which the name of Allah is mentioned?
He replied: It is permissible to enter the washroom with papers on which the name of Allah is mentioned, so long as they are in a pocket and are not out in the open; rather they are hidden and covered. End quote.
Fataawa min al-Tahaarah, p. 109
Based on that, it is permissible for you to enter the washroom with it, but the writing on the ring should be turned to the inside, so that it is covered and is not out in the open.
But it should be noted that if the ring is an engagement ring, it is not permissible for a Muslim to wear it, because that is an imitation of the Christians. This has been discussed previously in the answer to question no.21441.
And Allah knows best.