Is it valid to perform the prayer with the intention for the root of the prayer without specifying its time (such as Dhuhr or Asr)?

If the question is about the intention for Wudu (ablution), then specifying the prayer is not a condition; rather, the intention for Wudu for prayer or for removing minor impurity suffices. However, if the question is about the intention for Salah (prayer), then the prayer is not valid without it, and the prayer must be specified (Dhuhr, Asr, etc.) at the time of Takbirat al-Ihram (the opening Takbir). It is permissible to make the intention immediately before the prayer (as in intending Wudu and then praying immediately) according to the Hanafis, Hanbalis, and one of the opinions of the Malikis, whereas the Shafi'is and another opinion of the Malikis consider it invalid. As for making the intention much in advance, it invalidates the prayer by consensus.