Let $M=\bigoplus_{i \in \mathbb{Z}} \mathbb{C}e_i$ be an infinite dimensional $\mathbb{C}$-vector space, and let $\text{End}(M)$ denote the $\mathbb{C}$-algebra of $\mathbb{C}$-linear endomorphisms of $M$. Let $A$ and $B$ be two commuting elements in $\text{End}(M)$ satisfying the following condition: there exist integers $m \le n<0<p \le q$ satisfying $\text{gd}(-m,p)=\text{gcd}(-n,q)=1$, and such that for every $j \in \mathbb{Z}$, one has \[Ae_j=\sum_{i=j+m}^{j+n} a_{i,j}e_i, \quad \text{with } a_{i,j} \in \mathbb{C}, a_{j+m,j}a_{j+n,j} \ne 0,\] \[Be_j=\sum_{i=j+p}^{j+q} b_{i,j}e_i, \quad \text{with } b_{i,j} \in \mathbb{C}, b_{j+p,j}b_{j+q,j} \ne 0.\] Let $R \subset \text{End}(M)$ be the $\mathbb{C}$-subalgebra generated by $A$ and $B$. Note that $R$ is commutative and $M$ can be regarded as an $R$-module. (a) Show that $R$ is an integral domain and is isomorphic to $R \cong \mathbb{C}[x,y]/f(x,y)$, where $f(x,y)$ is a non-zero polynomial such that $f(A,B)=0$. (b) Let $K$ be the fractional field of $R$. Show that $M \otimes_R K$ is a $1$-dimensional vector space over $K$.

Let $n$ be a positive integer and $G$ be a finite group of order $n.$ A function $f:G \to G$ has the $(P)$ property if $f(xyz)=f(x)f(y)f(z)~\forall~x,y,z \in G.$ $\textbf{(a)}$ If $n$ is odd, prove that every function having the $(P)$ property is an endomorphism. $\textbf{(b)}$ If $n$ is even, is the conclusion from $\textbf{(a)}$ still true?

[b]a)[/b] Let $ u $ be a polynom in $ \mathbb{Q}[X] . $ Prove that the function $ E_u:\mathbb{Q}[X]\longrightarrow\mathbb{Q}[X] $ defined as $ E_u(P)=P(u) $ is an endomorphism. [b]b)[/b] Let $ E $ be an injective endomorphism of $ \mathbb{Q} [X] . $ Show that there exists a nonconstant polynom $ v $ in $ \mathbb{Q}[X] $ such that $ E(P)=P(v) , $ for any $ P $ in $ \mathbb{Q}[X] . $ [b]c)[/b] Let $ A $ be an automorphism of $ \mathbb{Q}[X] . $ Demonstrate that there is a nonzero constant polynom $ w $ in $ \mathbb{Q}[X] $ which has the property that $ A(P)=P(w) , $ for any $ P $ in $ \mathbb{Q}[X] . $ [i]Marcel Țena[/i]

Let be a group of order $ 2002 $ having the property that the application $ x\mapsto x^4 $ is and endomorphism of it. Show that this group is cyclic.

Let be a finite group $ G $ having an endomorphism $ \eta $ that has exactly one fixed point. [b]a)[/b] Demonstrate that the function $ f:G\longrightarrow G $ defined as $ f(x)=x^{-1}\cdot\eta (x) $ is bijective. [b]b)[/b] Show that $ G $ is commutative if the composition of the function $ f $ from [b]a)[/b] with itself is the identity function.

Let $ 0\neq\varrho\in\text{Hom}\left( \mathbb{Z}_4,\mathbb{Z}_2\right) ,$ $ \text{id}\neq\iota\in\text{Aut}\left( \mathbb{Z}_4\right) ,$ $ G:=\left\{ (x,y)\in\mathbb{Z}_4^2\big|x-y\in\ker\varrho\right\} , $ and $ \rho_1,\rho_2, $ the canonic projections of $ G $ into $ \mathbb{Z}_4. $ Prove that there exists an unique $ \nu\in\text{Hom}\left( \mathbb{Z}_4,G\right) $ such that $ \rho_1\circ\nu=\text{id} $ and $ \rho_2\circ\nu =\iota . $ Determine numerically this morphism.

Let $A$ be a commutative ring with $0 \neq 1$ such that for any $x \in A \setminus \{0\}$ there exist positive integers $m,n$ such that $(x^m+1)^n=x.$ Prove that any endomorphism of $A$ is an automorphism.

Let be a natural number $ n\ge 4, $ and a group $ G $ for which the applications $ \iota ,\eta : G\longrightarrow G $ defined by $ \iota (g) =g^n ,\eta (g) =g^{2n} $ are endomorphisms. Prove that $ G $ is commutative if $ \iota $ is injective or surjective. [i]Gh. Andrei[/i]