Is the husband entitled to dispose of a piece of gold agreed upon for the wife, which the husband had sent then retrieved and given to his mother? And in the event of the mother's death, is this piece inherited or returned to the wife who did not approve of the husband's action?

The dowry (Mahr) is a right of the wife. It is not permissible for the husband to take anything from it unless she willingly consents, as Allah Almighty says: "And give the women (upon marriage) their Mahr as a [willing] gift. But if they give up freely to you anything of it, then consume it with pleasure and ease." (Quran 4:4). And the Prophet, peace and blessings be upon him, said: "The property of a Muslim is not lawful except with his willing consent." Thus, the "Ta'liqa" (pendant/necklace), being part of the dowry, is the wife's right. The husband's disposal of it through a gift or sale is not valid without her permission. If the husband gave it to his mother, and then the mother died, it belongs to the wife, because it did not legally enter the mother's possession. The wife can forgive her husband or agree with him on compensation in its stead, or she can gift the "Ta'liqa" to his mother.