All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
It was authentically reported that Anas ibn Maalik narrated that the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, had a ring of silver, on which was engraved the words "Muhammad Rasool Allaah" and he used it as a seal. [Al-Bukhari, Muslim and others]
However, what you mentioned that the signet ring is in one of the museums and claimed to be the ring of the Prophet, sallallaahu ‘alayhi wa sallam, cannot be accepted for several reasons:
Firstly, this cannot be decisively proved.
Secondly, the ring of the Prophet, sallallaahu ‘alayhi wa sallam, was from silver not gold for Anas ibn Maalik narrated that the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, had a ring of silver. [Al-Bukhari]
Thirdly, that ring, i.e. the ring of the Prophet, sallallaahu ‘alayhi wa sallam, fell into a well as stated in the Hadeeth reported by Ibn ‘Umar who said: "The Prophet, sallallaahu ‘alayhi wa sallam, used a silver ring and placed its stone towards the palm of his hand and he engraved on it the words "Muhammad Rasool Allaah". He prohibited anyone to engrave (any other words) on it, and it was the ring which fell into the well of Arees. [Al-Bukhari]
It is authentically proved that the engraving of that ring was in three lines as narrated by Anas ibn Maalik who said: "The engraving of the ring of the Prophet, sallallaahu ‘alayhi wa sallam, was in three lines: 'Muhammad' in one line, 'Rasool' in another line, and 'Allaah' in another line." [At-Tirmithi]
Anas ibn Maalik narrated that when Abu Bakr became the Caliph, he wrote a letter to him (and stamped it with the Prophet's ring) and the engraving of the ring was in three lines: Muhammad in one line, 'Rasool' in another line, and 'Allaah' in another line. [Al-Bukhari]
Al-Haafith Ibn Hajar said in Al-Fat-h: "As for some scholars who said that the engraving was from bottom to top, i.e. Allaah was in the first line, Muhammad in the last one, I did not see any Hadeeth stating that; rather, the apparent meaning of the narration of Al-Ismaa‘eeli contradicts that. He said that Muhammad was in one line, then Rasool in the second line, and Allaah in the third line." [End quote]
However, if it was written from bottom to top, then this will imply no form of Shirk (polytheism).
It is more likely that the letters with which the engraving was written were in reverse so that the stamp on the letter would be legible (readable). Al-Haafith Ibn Hajar said: "It is also more likely that it was in that order; however, its writing was not in the normal way for the necessity of sealing a message requires that the letters engraved on the seal should be in reverse so that the stamp (on the letter) would be read correctly."
Allaah Knows best.