1. Two Maximal Ideals Proofs

Hello, I am stuck with these questions

1. Show that $\displaystyle \mathbb{Z} \oplus 2\mathbb{Z}$ is a maximal ideal in $\displaystyle \mathbb{Z} \oplus \mathbb{Z}$

2. Let $\displaystyle R$ be a commutative ring with unity. Prove that every prime ideal of $\displaystyle R$ is also a maximal ideal of $\displaystyle R$.

2. Hi

1) Assume some ideal $\displaystyle J$ of $\displaystyle \mathbb{Z}\oplus\mathbb{Z}$ strictly contains you ideal, i.e. there is an element in $\displaystyle J-\mathbb{Z}\oplus 2\mathbb{Z}$. Can you prove that the unity belongs to $\displaystyle J$ .

2) That's wrong. Take an integral domain which is not a field. Then $\displaystyle \{0\}$ is a prime ideal which is not maximal. A less extreme case: $\displaystyle (X) \subset \mathbb{Z}[X]$ is a prime ideal but not maximal (why?).

Were there other hypotheses or was it maximal implies prime (which is true)?

3. Gotta be a PID for 2) to be true.

But yeah for 1) you could just take that quotient and it is $\displaystyle \mathbb{Z}_2$ which is a field, so that ideal is maximal.