Problem 1.

Let $I$ and $J$ be ideals in a ring $R$. Prove that $I+J=\{a+b \mid a \in I, b \in J\}$ is an ideal.

Proof .

Problem 2.

Let $I$ and $J$ be ideals in a ring $R$. Show that $\{a b \mid a \in I, b \in J\}$ is not always an ideal with an example. Hint: Consider $I=J=(2, x)$ in $\mathbb{Z}[x]$.

Proof .

Remark：类似的例子是构造笛卡尔测度的时候，是用slice的sigma代数的生成元的笛卡尔积生成的最小sigma代数然后才是sigma代数封闭的，这在测度论里面也很重要，对应的是经典的单调类trick。

Problem 3.

Prove that if $I$ is a maximal ideal in a ring $R$ (meaning the only ideal which properly contains $I$ is $R$ ), then $R / I$ is a field. Hint: Use the Correspondence Theorem.

Proof .

Problem 4.

Let $\alpha \in \mathbb{C}$ be the real root of $x^{3}-2 x+5 .$ Express $(\alpha+1)\left(\alpha^{3}-2\right) \in \mathbb{Z}[\alpha]$
in terms of the basis $\left\{1, \alpha, \alpha^{2}\right\}$ for $\mathbb{Z}[\alpha]$ over $\mathbb{Z}$.

Problem 5.

Identify the ring $\mathbb{Z}[x] /\left(x-2, x^{2}+1\right)$.

##### Math 150B abstract agebra

Theory 太多 …Practice题目有点hold 不住？

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