What is the ruling on the the intention for ablution of someone who washes a part of their body for cooling or cleaning purposes?

Intention (Niyyah) is a pillar of ablution (Wudu') according to the majority of scholars. Thus, if one rejects the intention during their ablution, for example, by intending only to cool off or clean a limb when washing it, their ablution is not valid unless they return and wash it with the intention of ablution. However, if they intend both cooling off and ablution, their ablution is valid.