I am going through the following proof and I need some explanation
Thm: Let A be a commutative ring, an ideal is prime iff is an integral domain.
Assume that P is prime and pick suppose that is a zero in
we need to show that either
we have now so (how do we know that???) right? then by def of prime ideal thus no zero divisors and is int domain
suppose then as we have no zero divisors in we get so
why does implies that ? and
why does implies that ?
thanks for the explanaition