A set system $ (S,L)$ is called a Steiner triple system, if $ L\neq\emptyset$, any pair $ x,y\in S$, $ x\neq y$ of points lie on a unique line $ \ell\in L$, and every line $ \ell\in L$ contains exactly three points. Let $ (S,L)$ be a Steiner triple system, and let us denote by $ xy$ the thrid point on a line determined by the points $ x\neq y$. Let $ A$ be a group whose factor by its center $ C(A)$ is of prime power order. Let $ f,h: S\to A$ be maps, such that $ C(A)$ contains the range of $ f$, and the range of $ h$ generates $ A$. Show, that if \[ f(x) \equal{} h(x)h(y)h(x)h(xy)\] holds for all pairs $ x\neq y$ of points, then $ A$ is commutative, and there exists an element $ k\in A$, such that $ f(x) \equal{} kh(x)$ for all $ x\in S$.

Show that for every prime $p$ and integer $n$, there is an irreducible polynomial of degree $n$ in $\mathbb{Z}_p[x]$ and use that to show there is a field of size $p^n$.

Let $A$ be a ring. $a)$ Show that the set $Z(A)=\{a\in A|ax=xa,\ \text{for all}\ x\in A\}$ is a subring of the ring $A$. $b)$ Prove that, if any commutative subring of $A$ is a field, then $A$ is a field.

Prove that in an abelian ring $ A $ in which $ 1\neq 0, $ every element is idempotent if and only if the number of polynomial functions from $ A $ to $ A $ is equal to the square of the cardinal of $ A. $

Define the operator " $ * $ " on $ \mathbb{R} $ as $ x*y=x+y+xy. $ [b]a)[/b] Show that $ \mathbb{R}\setminus\{ -1\} , $ along with the operator above, is isomorphic with $ \mathbb{R}\setminus\{ 0\} , $ with the usual multiplication. [b]b)[/b] Determine all finite semigroups of $ \mathbb{R} $ under " $ *. $ " Which of them are groups? [b]c)[/b] Prove that if $ H\subset_{*}\mathbb{R} $ is a bounded semigroup, then $ H\subset [-2, 0]. $

In a lattice, connected the elements $ a \wedge b$ and $ a \vee b$ by an edge whenever $ a$ and $ b$ are incomparable. Prove that in the obtained graph every connected component is a sublattice. [i]M. Ajtai[/i]

Prove that the number of nilpotent elements of a commutative ring with an order greater than $ 8 $ and congruent to $ 3 $ modulo $ 6 $ is at most a third of the order of the ring.

We say that the rank of a group $ G$ is at most $ r$ if every subgroup of $ G$ can be generated by at most $ r$ elements. Prove that here exists an integer $ s$ such that for every finite group $ G$ of rank $ 2$ the commutator series of $ G$ has length less than $ s$. [i]J. Erdos[/i]

Prove that if an infinite, noncommutative group $ G$ contains a proper normal subgroup with a commutative factor group, then $ G$ also contains an infinite proper normal subgroup. [i]B. Csakany[/i]

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 $r,s,t$ positive integers which are relatively prime and $a,b \in G$, $G$ a commutative multiplicative group with unit element $e$, and $a^r=b^s=(ab)^t=e$. (a) Prove that $a=b=e$. (b) Does the same hold for a non-commutative group $G$?

For a distributive lattice $ L$, consider the following two statements: (A) Every ideal of $ L$ is the kernel of at least two different homomorphisms. (B) $ L$ contains no maximal ideal. Which one of these statements implies the other? (Every homomorphism $ \varphi$ of $ L$ induces an equivalence relation on $ L$: $ a \sim b$ if and only if $ a \varphi\equal{}b \varphi$. We do not consider two homomorphisms different if they imply the same equivalence relation.) [i]J. Varlet, E. Fried[/i]

Let $\mathcal K$ be a field with $2^{n}$ elements, $n \in \mathbb N^\ast$, and $f$ be the polynomial $X^{4}+X+1$. Prove that: (a) if $n$ is even, then $f$ is reducible in $\mathcal K[X]$; (b) if $n$ is odd, then $f$ is irreducible in $\mathcal K[X]$. [hide="Remark."]I saw the official solution and it wasn't that difficult, but I just couldn't solve this bloody problem.[/hide]

Let $ A $ be a ring in which $ 1\neq 0. $ If $ a,b\in A, $ then the following affirmations are equivalent: $ \text{(i)}\quad aba=a\wedge ba^2b=1 $ $ \text{(ii)}\quad ab=ba=1 $ $ \text{(iii)}\quad \exists !b\in A\quad aba=a $

Let $A$ be a finite ring. Show that there exists two natural numbers $m,p$ where $m> p\ge 1$, such that $a^m=a^p$ for all $a\in A$.

Let $A$ be Abelian group of order $p^4$, where $p$ is a prime number, and which has a subgroup $N$ with order $p$ such that $A/N\approx\mathbb{Z}/p^3\mathbb{Z}$. Find all $A$ expect isomorphic.

Prove that the intersection of all maximal left ideals of a ring is a (two-sided) ideal.

Let $ K$ be a subset of a group $ G$ that is not a union of lift cosets of a proper subgroup. Prove that if $ G$ is a torsion group or if $ K$ is a finite set, then the subset \[ \bigcap _{k \in K} k^{-1}K\] consists of the identity alone. [i]L. Redei[/i]

For a finite group $G$ we denote by $n(G)$ the number of elements of the group and by $s(G)$ the number of subgroups of it. Decide whether the following statements are true or false. a) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}<a.$ b) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}>a.$

Let $ R$ be an Artinian ring with unity. Suppose that every idempotent element of $ R$ commutes with every element of $ R$ whose square is $ 0$. Suppose $ R$ is the sum of the ideals $ A$ and $ B$. Prove that $ AB\equal{}BA$. [i]A. Kertesz[/i]

[color=darkred] Let $(R,+,\cdot)$ be a ring and let $f$ be a surjective endomorphism of $R$ such that $[x,f(x)]=0$ for any $x\in R$ , where $[a,b]=ab-ba$ , $a,b\in R$ . Prove that: [list] [b]a)[/b] $[x,f(y)]=[f(x),y]$ and $x[x,y]=f(x)[x,y]$ , for any $x,y\in R\ ;$ [b]b)[/b] If $R$ is a division ring and $f$ is different from the identity function, then $R$ is commutative. [/list] [/color]

Let $A,B\in M_2(\mathbb Z)$ s.t. $AB=\begin{pmatrix}1&2005\\0&1\end{pmatrix}$. Prove that there is a matrix $C\in M_2(\mathbb Z)$ s.t. $BA=C^{2005}$. [i]Dinu Serbanescu[/i]

Given a ring $\left( A,+,\cdot \right)$ that meets both of the following conditions: (1) $A$ is not a field, and (2) For every non-invertible element $x$ of $ A$, there is an integer $m>1$ (depending on $x$) such that $x=x^2+x^3+\ldots+x^{2^m}$. Show that (a) $x+x=0$ for every $x \in A$, and (b) $x^2=x$ for every non-invertible $x\in A$.

Let $G$ be a group with $m$ elements and let $H$ be a proper subgroup of $G$ with $n$ elements. For each $x\in G$ we denote $H^x = \{ xhx^{-1} \mid h \in H \}$ and we suppose that $H^x \cap H = \{e\}$, for all $x\in G - H$ (where by $e$ we denoted the neutral element of the group $G$). a) Prove that $H^x=H^y$ if and only if $x^{-1}y \in H$; b) Find the number of elements of the set $\bigcup_{x\in G} H^x$ as a function of $m$ and $n$. [i]Calin Popescu[/i]

Problem 2. A group $\left( G,\cdot \right)$ has the propriety$\left( P \right)$, if, for any automorphism f for G,there are two automorphisms g and h in G, so that $f\left( x \right)=g\left( x \right)\cdot h\left( x \right)$, whatever $x\in G$would be. Prove that: (a) Every group which the property $\left( P \right)$ is comutative. (b) Every commutative finite group of odd order doesn’t have the $\left( P \right)$ property. (c) No finite group of order $4n+2,n\in \mathbb{N}$, doesn’t have the $\left( P \right)$property. (The order of a finite group is the number of elements of that group).