\(\Omega_\mathbb{B_p} \subseteq \Omega_\mathbb{B_u} \subseteq \Omega_\mathbb{B_b} \) 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.