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 $A=(a_{ij})$ be an $m\times n$ real matrix with at least one non-zero element. For each $i\in\{1,\ldots,m\}$, let $R_i=\sum_{j=1}^na_{ij}$ be the sum of the $i$-th row of the matrix $A$, and for each $j\in\{1,\ldots,n\}$, let $C_j =\sum_{i=1}^ma_{ij}$ be the sum of the $j$-th column of the matrix $A$. Prove that there exist indices $k\in\{1,\ldots,m\}$ and $l\in\{1,\ldots,n\}$ such that $$a_{kl}>0,\qquad R_k\ge0,\qquad C_l\ge0,$$or $$a_{kl}<0,\qquad R_k\le0,\qquad C_l\le0.$$

Call a polynomial positive reducible if it can be written as a product of two nonconstant polynomials with positive real coefficients. Let $ f(x)$ be a polynomial with $ f(0)\not\equal{}0$ such that $ f(x^n)$ is positive reducible for some natural number $ n$. Prove that $ f(x)$ itself is positive reducible. [L. Redei]

Let $ p\geq 7$ be a prime number, $ \zeta$ a primitive $ p$th root of unity, $ c$ a rational number. Prove that in the additive group generated by the numbers $ 1,\zeta,\zeta^2,\zeta^3\plus{}\zeta^{\minus{}3}$ there are only finitely many elements whose norm is equal to $ c$. (The norm is in the $ p$th cyclotomic field.) [i]K. Gyory[/i]

Let $F$ be a field whose characteristic is not $2$, let $F^*=F\setminus\left\{0\right\}$ be its multiplicative group and let $T$ be the subgroup of $F^*$ constituted by its finite order elements. Prove that if $T$ is finite, then $T$ is cyclic and its order is even.

Let $\left( A,+, \cdot \right)$ be a ring that verifies the following properties: (i) it has a unit, $1$, and its order is $p$, a prime number; (ii) there is $B \subset A, \, |B| = p$, such that: for all $x,y \in A$, there is $b \in B$ such that $xy = byx$. Prove that $A$ is commutative. [i]Ion Savu[/i]

Let $ G$ be anon-commutative group. Consider all the one-to-one mappings $ a\rightarrow a'$ of $ G$ onto itself such that $ (ab)'\equal{}b'a'$ (i.e. the anti-automorphisms of $ G$). Prove that this mappings together with the automorphisms of $ G$ constitute a group which contains the group of the automorphisms of $ G$ as direct factor.

Let $ \mathcal G$ be the set of all finite groups with at least two elements. a) Prove that if $ G\in \mathcal G$, then the number of morphisms $ f: G\to G$ is at most $ \sqrt [p]{n^n}$, where $ p$ is the largest prime divisor of $ n$, and $ n$ is the number of elements in $ G$. b) Find all the groups in $ \mathcal G$ for which the inequality at point a) is an equality.

Let $ G $ be the set of matrices of the form $ \begin{pmatrix} a&b\\0&1 \end{pmatrix} , $ with $ a,b\in\mathbb{Z}_7,a\neq 0. $ [b]a)[/b] Verify that $ G $ is a group. [b]b)[/b] Show that $ \text{Hom}\left( (G,\cdot) ; \left( \mathbb{Z}_7,+ \right) \right) =\{ 0\} $

Suppose there exists a group with exactly $n$ subgroups of index 2. Prove that there exists a finite abelian group $G$ that has exactly $n$ subgroups of index 2.

Let be a ring that has the property that all its elements are the product of two idempotent elements of it. Show that: [b]a)[/b] $ 1 $ is the only unit of this ring. [b]b)[/b] this ring is Boolean.

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.

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 $G$ be a group and let $a$ be a constant member of it. Prove that \[G_a = \{x | \exists n \in \mathbb Z , x=a^n\}\] Is a subgroup of $G.$

Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\ $\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\ $\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\ $\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true? \\ \\ [i] (Bogdan Blaga)[/i]

Let $\displaystyle \mathcal K$ be a finite field. Prove that the following statements are equivalent: (a) $\displaystyle 1+1=0$; (b) for all $\displaystyle f \in \mathcal K \left[ X \right]$ with $\displaystyle \textrm{deg} \, f \geq 1$, $\displaystyle f \left( X^2 \right)$ is reducible.

Let $p$ be a prime number, $n$ a natural number which is not divisible by $p$, and $\mathbb{K}$ is a finite field, with $char(K) = p, |K| = p^n, 1_{\mathbb{K}}$ unity element and $\widehat{0} = 0_{\mathbb{K}}.$ For every $m \in \mathbb{N}^{*}$ we note $ \widehat{m} = \underbrace{1_{\mathbb{K}} + 1_{\mathbb{K}} + \ldots + 1_{\mathbb{K}}}_{m \text{ times}} $ and define the polynomial \[ f_m = \sum_{k = 0}^{m} (-1)^{m - k} \widehat{\binom{m}{k}} X^{p^k} \in \mathbb{K}[X]. \] a) Show that roots of $f_1$ are $ \left\{ \widehat{k} | k \in \{0,1,2, \ldots , p - 1 \} \right\}$. b) Let $m \in \mathbb{N}^{*}.$ Determine the set of roots from $\mathbb{K}$ of polynomial $f_{m}.$

Let $A$ be a finite ring and $a,b \in A,$ such that $(ab-1)b=0.$ Prove that $b(ab-1)=0.$

Let $ p>5$ be a prime number. Prove that every algebraic integer of the $ p$th cyclotomic field can be represented as a sum of (finitely many) distinct units of the ring of algebraic integers of the field. [i]K. Gyory[/i]

Determine all non-trivial finite rings with am unit element in which the sum of all elements is invertible. [i]Mihai Opincariu[/i]

Prove that the group morphisms $f: (\mathbb{C},+)\to(\mathbb{C},+)$ for which there exists a positive $\lambda$ such that $|f(z)| \leq \lambda |z|$ for all $z\in\mathbb{C}$, have the form \[ f(z) = \alpha z + \beta \overline{z} \] for some complex $\alpha$, $\beta$. [i]Cristinel Mortici[/i]

Show that there is an unique group $ G $ (up to isomorphism) of order $ 1986 $ which has the property that there is at most one subgroup of it having order $ n, $ for every natural number $ n. $

$G$ is a group that order of each element of it Commutator group is finite. Prove that subset of all elemets of $G$ which have finite order is a subgroup og $G$.

For any $k \in \mathbb{Z},$ define $$F_k=X^4+2(1-k)X^2+(1+k)^2.$$ Find all values $k \in \mathbb{Z}$ such that $F_k$ is irreducible over $\mathbb{Z}$ and reducible over $\mathbb{Z}_p,$ for any prime $p.$ [i]Marius Vladoiu[/i]

[b]a)[/b] Show that, for any distinct natural numbers $ m,n, $ the rings $ \mathbb{Z}_2\times \underbrace{\cdots}_{m\text{ times}} \times\mathbb{Z}_2,\mathbb{Z}_2\times \underbrace{\cdots}_{n\text{ times}} \times\mathbb{Z}_2 $ are homomorphic, but not isomorphic. [b]b)[/b] Show that there are infinitely many pairwise nonhomomorphic rings of same order.