Must the questioner leave his mother to live alone and rent a house for himself to live in with his wife, or divorce his wife without a valid reason, given that quarrels between the mother and the wife recur?

It is the wife's right to have a separate dwelling, and she is not obligated to live with the husband's family. If this separate dwelling becomes available, the problem might be resolved. If the wife is righteous, do not divorce her, even if the mother requests it. If a separate dwelling cannot be provided and the wife's family agrees to your staying with them, then be patient with that.