An interesting way to think about the forward direction in Problem 5: If n is not a prime, then we can find 1 < a < n such that a divides n. If we consider the ideal generated by a, then it only contains multiples of a and as 1 < a < n, this ideal does NOT contain all of 1 to n. So this ideal is different from both the zero ideal and the unit ideal(the full ring) and hence one can use Problem 4 now to deduce that Z/nZ is not a field!
sir I have a doubt, in problem 4 you said that Kernel cannot be R because 1 goes to 1, but what if all elements in R go to the zero element in R', it's still a ring homomorphism and it's not injective(assuming R' has some elements other than zero), why can this case not happen?
He said in the beginning that we need to assume R' is not the zero ring, so that 1 and 0 are different in R'. So the zero map, which maps everything in R to R' is not a homomorphism because 1 in R has to be mapped to 1 in R' which is different from 0. So you cannot consider the zero map as a homomorphism as R' is assumed to be not the zero ring
An interesting way to think about the forward direction in Problem 5: If n is not a prime, then we can find 1 < a < n such that a divides n. If we consider the ideal generated by a, then it only contains multiples of a and as 1 < a < n, this ideal does NOT contain all of 1 to n. So this ideal is different from both the zero ideal and the unit ideal(the full ring) and hence one can use Problem 4 now to deduce that Z/nZ is not a field!
good work
Thank you Sir 🙏
sir I have a doubt, in problem 4 you said that Kernel cannot be R because 1 goes to 1, but what if all elements in R go to the zero element in R', it's still a ring homomorphism and it's not injective(assuming R' has some elements other than zero), why can this case not happen?
He said in the beginning that we need to assume R' is not the zero ring, so that 1 and 0 are different in R'. So the zero map, which maps everything in R to R' is not a homomorphism because 1 in R has to be mapped to 1 in R' which is different from 0. So you cannot consider the zero map as a homomorphism as R' is assumed to be not the zero ring