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.

Prove that the direct product of a finite number of R-free modules is an R-free module.

Show that every ideal of $\mathbb{Z}$ is a free $\mathbb{Z}$-module .

Let $R$ be a commutative ring. Prove that $HOM_R(R,M)$ and $M$ are isomorphic as left $R$-modules

Let $R$ be a ring. Give an example of a map from one $R$-module to another which is a group homomorphism but not an $R$-module homomorphism.