If someone prays while knowing that they are not in a state of ritual purity (wudu'), does that invalidate their fast?

There is no relationship between intentionally performing prayer without Wudu (ablution) and the invalidation of the fast, unless it is asserted that the one who does so has committed an act of disbelief (kufr), which is a view held by some scholars. However, the majority of scholars believe that whoever prays without purification, knowing and intending to do so, has committed a grave sin but does not commit an act of disbelief. This is the authoritative fatwa (definition). Based on the preponderant opinion, the fast of this person is valid, and what is obligatory upon him is to repent to Allah the Almighty.