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.

An element \(m\) of the \(R\)-module \(M\) is called a torsion element if \(rm=0\) for some non-zero element \(r \in R\) .

(i) Prove that if \(R\) is an integral domain then \(T\) or \((M)\) , the set of torsion elements of \(M\) , is a submodule of \(M\) (called the torsion submodule of \(M\)).

(ii) Give an example of a ring \(R\) and an \(R\)-module \(M\) such that \(T\) or\((M)\) is not an \(R\)-submodule.

(iii) If \(R\) has zero-divisors, show that every non-zero \(R\)-module has non-zero torsion elements.

An \(R\)-module \(M\) is called a torsion module if for each \(m \in M\) there is a non-zero element \(r \in R\) such that \(rm =0\), where r may depend on \(m\). Prove that every finite abelian group is a torsion \(\mathbb{Z}\)-module. Give an example of an infinite abelian group that is a torsion \(\mathbb{Z}\)-module.