# Math Help - Ideal

1. ## Ideal

Consider the ring Z of integers.

(a) Prove that the ideal I = (4) is not a maximal ideal.
Hint. Find an ideal J of Z such that (4) is a subset of J subset of Z but J does not equal (4) and J does not equal Z.

(b) Prove that the ideal I = (3) is a maximal ideal.
Hint. Suppose that J is any ideal of Z such that (3) is a subset of J subset of Z. Prove that J = I or J = Z.

2. The hints outline exactly what you need to do. Could you explain what trouble you are having?

3. I know that an ideal I of a ring R is said to be a maximal ideal if I does not equal R and for every ideal J of R,
if I is a subset of J subset of R then either J = I or J = R.
Therefore, there is no ideal strictly between I and R.

4. So if J = (2) would that hold true in the "hint." Also, how do I then link that back to what I want to prove. Would I do a contradiction?