To adjust her status, you would file an I-130 and I-485 (plus other forms) so she can stay and get a green card. She must remain in the US until she gets AP or a green card which means 5-6 months before she can leave the US. If she leaves the US before getting AP, then she abandons her adjustment of status.
If you want her to go back to the DR next week, then file just the I-130 for consular processing for an immigration visa. This would not require her to adjust her status.
You will need the mother's written permission for her to immigrate.
