Math 120B HW 7

Due Tuesday, May 19.
  1. Let $R$ be a commutative ring with unity and let $a \in R$. Define $aR = \{ar : r \in R\}$.

    1. Prove that $aR$ is an ideal of $R$.

    2. Prove that $aR$ is the smallest ideal of $R$ containing $a$, in the following sense:

      $a \in aR$, and if $N$ is any other ideal of $R$ such that $a \in N$, then $aR \subseteq N$.

    The ideal $aR$ is called the principal ideal of $R$ generated by $a$. It is also denoted by $\langle a \rangle$. Warning: it is not always the same as the cyclic subgroup of $(R,+)$ generated by $a$.
  2. Let $N$ be the principal ideal of $\mathbb{Q}[x]$ generated by $x^2 - 1$. Is there a homomorphism from $\mathbb{Q}[x] \to \mathbb{R}$ whose kernel is $N$? Justify your answer.

  3. Let $R_1$ and $R_2$ be rings. Let $N_1$ be an ideal of $R_1$ and let $N_2$ be an ideal of $R_2$. Then $N_1 \times N_2$ is an ideal of $R_1 \times R_2$. (I don't ask you to write a proof of this, but you should be able to do so.)

    Define an isomorphism $\phi : (R_1 \times R_2)/(N_1 \times N_2) \to (R_1 / N_1) \times (R_2 / N_2)$, and prove that your definition defines a function, i.e. "$\phi$ is well-defined." (I don't ask you to write the rest of a proof that your function is an isomorphism, but you should be able to do so.)

  4. For each of the following ideals of $\mathbb{Z} \times \mathbb{Z}$, say whether or not it is prime. Justify any "no" answers.

    1. $2 \mathbb{Z} \times 3 \mathbb{Z}$
    2. $\{0\} \times \{0\}$
    3. $\mathbb{Z} \times \mathbb{Z}$
    4. $\mathbb{Z} \times 3 \mathbb{Z}$
    5. $\mathbb{Z} \times 4 \mathbb{Z}$
    6. $\{0\} \times \mathbb{Z}$.
  5. For each of the ideals of $\mathbb{Z} \times \mathbb{Z}$ listed in problem 4, say whether or not it is maximal. Justify any "no" answers with an example of an ideal in between the given ideal and $\mathbb{Z} \times \mathbb{Z}$.

  6. Let $R$ be a commutative ring with unity $1 \ne 0$.

    (Note that every maximal ideal of $R$ is prime, by the following argument: "$N$ is maximal" $\implies$ "$R/N$ is a field" $\implies$ "$R/N$ is an integral domain" $\implies$ "$N$ is prime." This problem asks you to prove that every maximal ideal of $R$ is prime by a more direct argument, without using factor rings.)

    Let $N$ be a maximal ideal of $R$ and let $a,b \in R$ with $ab \in N$ and $a \notin N$. To prove that $N$ is prime, we want to show that $b \in N$.

    Consider the ideal $A = \{x + ar : x \in N \text{ and } r \in R\}$. (The proof that $A$ is an ideal of $R$ is similar to problem 1, so I don't ask you to write it.)

    1. Prove that $1 \in A$.

    2. Use the previous part to prove that $b \in N$.