Several legends explain how Ganesh broke his right tusk, which gives him the name of Ekadanta "The Lord who has only one tusk".
The first legend (in the Brahmanda-Purana) is related to a battle between Ganesh and Parashurâma. Parashurama was one of the Vishnu incarnations (avatara), born on earth to teach wisdom to the governing class, the Kshatriya , who had become arrogant and oppressed people. Parashurama meditated on Shiva and got the divine axe, Parashu. This axe helped him to fight against all the corrupted princes, inspired by devils.
Deeply grateful to Shiva, he went to Mount Kailash to bow to his guru. But Ganesh, who was guarding the entrance of the palace, did not allow him to proceed. Ganesh told him to wait for the Shiva permission.
Parashurâma thought : "I am a Shiva devotee, such a rule cannot be applied to me".
Ganesh persisting to bar the way, Parashurama, usually peppery, stroke violently the Ganesh tusk with his axe and broke it.
Then Shiva and Parvati arrived and blamed Parashurâma who bowed down before Ganesh and supplicated to obtain his forgiveness and blessing. Then Ganesh was named Ekadanta "The Lord with one tusk".
According to another legend, Ganesh broke himself his tusk during the battle against Gajamukhâsura (the elephant-headed Asura ). Taking the advice of Shukracharya, the Asura guru, this demon followed severe penances. Thus, he got unconquerable powers from Shiva. But he misused those powers to harass the gods who went to Ganesh and requested his help.
Ganesh did not hesitate to give battle to this demon. During the fight, he understood that the demon could not defeated, because of his particular powers. Then, Ganesh broke his right tusk and threw it to Gajamukhâsura. He pursued him and converted him in a mouse. Then he rode this mouse, which he used as a mount, keeping it under control.
According to another Purana story, the Ganesh rat was actually the Gandharva Krauncha. One day, at the Indra Court, Krauncha insulted the Sage Vâmadeva who revenged himself, making him a big rat. This rat, as all the rats do, went in the ashram of the Sage Parâchara and caused a lot of damages in the house. The Rishi invoked Vinayaka (an other name for Ganesh) to safeguard his modest dwelling. Ganesh appeared, rode the rat as his vehicle and mastered it.
A Purana legend imputes the loss of the tusk to a fight between Ganesh and Shiva himself.
Finally, there is the story between Ganesh and the Moon narrated above.
Whatever the version of these puranic stories, Ganesh chose the rat as a vehicle for an obvious reason : this animal is really a detrimental one and Ganesh was able to keep it under his strict control.