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.

Let $R$ be an integral domain and let $x \in R \backslash\{0\}$ . Show that $R \cong Rx$ as R-submodules. But $R \cong Rx$ as rings if and only if $x$ is a unit in $R$.

Let $R$ be a ring. Show that an $R$-module $M$ is a simple module if and only if $M \cong R \backslash I$ for some maximal left ideal $I$ of $R$ .

Let $R$ be a commutative ring with identity, and let $e \neq 0,1$  be an idempotent in $R$. Prove that $Re$ cannot be a free $R$-module.