The IRS does not consider your situation to be a "rental" if you are renting from a family member at below fair market value. Thus from a tax perspective no formal rental agreement is necessary.
Also, sharing the cost of goods or services from which you yourself benefit does not constitute a gift.
If this is merely a cost sharing arrangement where the amount paid is below fair market rental, there would be no reportable income to you. If the “rent” amount is fair market value, or more, there is still some question as to whether you even have to report it, as it almost always comes out zero. Most people take the attitude that it is not income; it's just room mates sharing expenses and ignore it. Family, as opposed to unrelated roommates, makes that position stronger.
What your mother is not allowed to do, because it is her own home (she has "personal use") is claim a loss from this activity, to offset other income. Because of the "personal use rule", her deductions are limited to her income. Net effect ZERO.
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