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.

Let $R$ be an integral domain. Prove that every finitely generated torsion $R$-module has a non-zero annihilator. Give an example of a torsion $R$-module whose annihilator is the zero ideal

Let $K/F$ be an extension of fields. Let $\alpha ,\beta \in K$ be algebraic over $F$ and the degrees of the minimal polynomials of $\alpha$ and $\beta$ over $F$ are $m$ and $n$, respectively. Show that if $(m,n) =1$, then $[F(\alpha ,\beta ):F] =mn$ .

Let $K/F$ be an extension of fields. Show that if every ring $R$ with $F \subseteq R \subseteq K$ is a field, then $K$ is an algebraic extension of $F$ .

Find the degree and a basis for $\mathbb{Q}( \sqrt{2} , \sqrt{3} )$ over $\mathbb{Q}$ .Also show that $\mathbb{Q}( \sqrt{2} , \sqrt{3} ) = \mathbb{Q}( \sqrt{2} + \sqrt{3} )$

Find the degree and a basis for $\mathbb{Q} ( \sqrt{3} + \sqrt{5} )$ over $\mathbb{Q} ( \sqrt{15})$ .

Prove that $[ \mathbb{Q} ( \sqrt {3-4i} \; + \; \sqrt{3 + 4i} ) : \mathbb{Q} ]=1$ .

Determine the degree of the extension  $\mathbb{Q} \large ( \sqrt {1 +\sqrt{-3}} \; + \sqrt{1- \sqrt{-3}} )$ over $\mathbb{Q}$ .