Is my mother required by law to submit my deceased fathers will to probate court in Texas?

Texas law requires that a Will be submitted to probate.  Please note that if your father left so little property that selling it would not pay for the cost of probate, his final expenses and his funeral, most lawyers will probably advise against doing so.  Please also note that most spouses leave everything to each other, which would be the result if there were no Will and the deceased had no children who were not the children of the surviving spouse.