数学代写|Math 150B abstract agebra homework5|抽象代数代写

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.

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 $\mathbb{Z}[x]$.

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.

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 $\mathbb{Z}$.

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