$\Omega_\mathbb{B_p} \subseteq \Omega_\mathbb{B_u} \subseteq \Omega_\mathbb{B_b}$ with strict inclusions if $I$ is infinite.

Let $X$ and $Y$ be compact spaces. Assume further that $Y$ is Hausdorff. Let $f:X \rightarrow Y$ be a continuous surjection. Define the equivalence relation $\sim$ on $X$ by $x_1 \sim x_2$ iff $f(x_1)=f(x_2)$. Then $g:X / \sim \rightarrow Y$ given by $g(|x|)=f(x)$ is a well-defined homeomorphism.

Show that in the ring $\mathbb{Z}[\sqrt{-3}]$ the gcd of $4$ and $2+2\sqrt{-3}$ does not exist.

Describe the field of quotients of  $\mathbb{Z} [\sqrt{n}]$ , where $n$ is a square-free integer.

Show that the domains $\mathbb{Z} [\sqrt{-6}]$ and $\mathbb{Z} [\sqrt{-7}]$ are not UFDs.

Let $R$ be an integral domain that is not a field; show that the polynomial ring $R[x]$ is not a PID.

Show that the polynomial ring $F[x,y]$ in two variables over a field $F$ is a UFD but not a PID.

Let $R$ be a commutative ring with identity and $f(x) \in r[x]$ . Show that an element $a \in R$  is a multiple root of $f(x)$ if and only if a is a root of $f^\prime(x)$, where $f^\prime(x)$ is the (formal) derivative of $f(x)$ .

Let $F$ be a field and $f(x) \in F[x]$] be a polynomial of degree 2 or 3. Show that $f(x)$ is irreducible over $F$ if and only if $f(x)$ has no root in $F$. Give an example to show that the same is not true if deg $f(x) \ge 4.$

Show that the polynomials $2x^4 +6x^3-9x^2+15$ and $x^6+x^3+1$ are irreducible over $\mathbb{Z}$ .

Let $M$ be an $R$-module and $x \in M$ be such that $rx=0$ for any $r \in R$ implies that $r=0$ . Then show that $R_x \cong R$ as $R$-modules.