R is a ring. So we have at our disposal just the axioms which make a set qualify as ring.
Q1. Is it true that a.b = 0 => b.a = 0? Personally I don't think so - but couldn't think of a quick counter example.
Q2. Assuming R has unit element, 1. Is it true that a.b = 1 => b.a = 1? My hunch again is that not necessarily.