Can I petition my mother ?she live here for 10 years and then she left the country for almost 6 yrs now and her visa expired,

A US citizen child over 21 years, can sponsor his or her mother for a green card, provided that she entered the United States legally even though she is out of status. However if she exited the United States after overstaying for more than one year, then she is barred from returning for a period of 10 years, unless you can qualify for a waiver.