My mom has a rash that is burning and it's starting from her underarm all the way up the back of her arm. no fever. it's red with pus

It's possible that: ...your mom has shingles (Herpes Zoster) for which medications can be prescribed that can help with her symptoms and prevent a condition called "post-herpetic neuralgia." In any case, she should see a doctor about this. It's a fairly common condition that a primary care physician (general internist or family doctor) can handle. If they can't, they can refer her to a dermatologist.