Let $(A,+,\cdot)$ be an unit ring with the property: for all $x\in A$, \[ x+x^{2}+x^{3}=x^{4}+x^{5}+x^{6} \] [list=a] [*]Let $x\in A$ and let $n\geq2$ be an integer such that $x^{n}=0$. Prove that $x=0$. [*]Prove that $x^{4}=x$, for all $x\in A$.[/list]

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?

Let $n$ and $q$ be two natural numbers, $n\ge 2$, $q\ge 2$ and $q\not\equiv 1 (\text{mod}\ 4)$ and let $K$ be a finite field which has exactly $q$ elements. Show that for any element $a$ from $K$, there exist $x$ and $y$ in $K$ such that $a = x^{2^n} + y^{2^n}$. (Every finite field is commutative).

Let $ A$ and $ B$ be two Abelian groups, and define the sum of two homomorphisms $ \eta$ and $ \chi$ from $ A$ to $ B$ by \[ a( \eta\plus{}\chi)\equal{}a\eta\plus{}a\chi \;\textrm{for all}\ \;a \in A\ .\] With this addition, the set of homomorphisms from $ A$ to $ B$ forms an Abelian group $ H$. Suppose now that $ A$ is a $ p$-group ( $ p$ a prime number). Prove that in this case $ H$ becomes a topological group under the topology defined by taking the subgroups $ p^kH \;(k\equal{}1,2,...)$ as a neighborhood base of $ 0$. Prove that $ H$ is complete in this topology and that every connected component of $ H$ consists of a single element. When is $ H$ compact in this topology? [L. Fuchs]

Define functions $f$ and $g$ as nonconstant, differentiable, real-valued functions on $R$. If $f(x+y)=f(x)f(y)-g(x)g(y)$, $g(x+y)=f(x)g(y)+g(x)f(y)$, and $f'(0)=0$, prove that $\left(f(x)\right)^2+\left(g(x)\right)^2=1$ for all $x$.

Let $ A $ be a ring, $ a\in A, $ and let $ n,k\ge 2 $ be two natural numbers such that $ n\vdots\text{char} (A) $ and $ 1+a=a^k. $ Show that the following propositions are true: [b]a)[/b] $ \forall s\in\mathbb{N}\quad \exists p_0,p_1,\ldots ,p_{k-1}\in\mathbb{Z}_{\ge 0}\quad a^s=\sum_{i=0}^{k-1} p_ia^{i} . $ [b]b)[/b] $ \text{ord} (a)\neq\infty . $

Let $K$ be a finite field with $q$ elements and let $n \ge q$ be an integer. Find the probability that by choosing an $n$-th degree polynomial with coefficients in $K,$ it doesn't have any root in $K.$

Given $n\ge 2$ a natural number, $(K,+,\cdot )$ a body with commutative property that $\underbrace{1+...+}_{m}1\ne 0,m=2,...,n,f\in K[X]$ a polynomial of degree $n$ and $G$ a subgroup of the additive group $(K,+,\cdot )$, $G\ne K.$Show that there is $a\in K$ so$f(a)\notin G$.

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Let $H \leq G, |H | = 3.$ What can be said about $|N_{G}(H ) : C_{G}(H )|$?

Suppose that the automorphism group of the finite undirected graph $ X\equal{}(P, E)$ is isomorphic to the quaternion group (of order $ 8$). Prove that the adjacency matrix of $ X$ has an eigenvalue of multiplicity at least $ 4$. ($ P\equal{} \{ 1,2,\ldots, n \}$ is the set of vertices of the graph $ X$. The set of edges $ E$ is a subset of the set of all unordered pairs of elements of $ P$. The group of automorphisms of $ X$ consists of those permutations of $ P$ that map edges to edges. The adjacency matrix $ M\equal{}[m_{ij}]$ is the $ n \times n$ matrix defined by $ m_{ij}\equal{}1$ if $ \{ i,j \} \in E$ and $ m_{i,j}\equal{}0$ otherwise.) [i]L. Babai[/i]

Let $ R$ be a ring with 1. Every element in $ R$ can be written as product of idempotent ($ u^n\equal{}u$ for some $ n$) elements. Prove that $ R$ is commutative

Let $K$ commutative field with $8$ elements. Prove that $(\exists)a\in K$ such that $a^3=a+1$. [i]Mircea Becheanu[/i]

Consider the endomorphism ring of an Abelian torsion-free (resp. torsion) group $ G$. Prove that this ring is Neumann-regular if and only if $ G$ is a discrete direct sum of groups isomorphic to the additive group of the rationals (resp. ,a discrete direct sum of cyclic groups of prime order). (A ring $ R$ is called Neumann-regular if for every $ \alpha \in R$ there exists a $ \beta \in R$ such that $ \alpha \beta \alpha\equal{}\alpha$.) [i]E. Freid[/i]

Let $p$ be a prime number. Call a positive integer $n$ interesting if \[x^n-1=(x^p-x+1)f(x)+pg(x)\] for some polynomials $f$ and $g$ with integer coefficients. a) Prove that the number $p^p-1$ is interesting. b) For which $p$ is $p^p-1$ the minimal interesting number?

Problem 4. Let$\left( A,+,\cdot \right)$ be a ring with the property that $x=0$ is the only solution of the ${{x}^{2}}=0,x\in A$ecuation. Let $B=\left\{ a\in A|{{a}^{2}}=1 \right\}$. Prove that: (a) $ab-ba=bab-a$, whatever would be $a\in A$ and $b\in B$. (b) $\left( B,\cdot \right)$ is a group

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Let $H \leq G$ and $a, b \in G.$ Prove that $|aH \cap Hb|$ is either zero or a divisor of $|H |.$