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.