Let be a ring and . Suppose that where and are idempotent elements and . Let and . If and then prove that is a division ring .
first you need to clarify two thing for us:
1) for noncommutative rings, there are more than one nilradical. so you need to give us the definition of in here.
2) is supposed to mean that has only one maximal left ideral? (this is the first time i see this notation!)
for now, just see that are central, i.e. they are in the center of and thus, since we have