I know that Z/pZ is a field therefore pZ must be a maximal ideal because of the theorem that states "R/I is a field if and only if I is a maximal ideal" but I want to see a direct proof of it because I hope it would give me an idea how to prove something is a maximal ideal in a general field. Suppose that I've been given an Ideal in a ring R, how can I prove that it's maximal? What are the general strategies that I should consider first? (except the definition of a maximal ideal of course!). I'd like to see a direct proof of pZ being a maximal ideal and also learn how to deal with the situation in a general case. Thanks in advance.