Let $R$ be a commutative ring with unity and $U\left(R\right)$ be the set of unit elements of $R$. In this paper, we introduce and investigate some properties of a new kind of graph on the ring $R$, namely, the prime ideals intersection graph of $R$, denoted by $G_p\left(R\right)$. The $G_p\left(R\right)$ is a graph with vertex set $R^{*}-U\left(R\right)$ and two distinct vertices $a$ and $b$ are adjacent if and only if there exists a prime ideal $𝔭$ of $R$ such that $a,b\in 𝔭$. We obtain necessary and sufficient conditions on $R$ such that $G_p\left(R\right)$ is disconnected. We find the diameter and...

Let $G$ be a finite group. The prime graph of $G$ is a simple graph $\Gamma \left(G\right)$ whose vertex set is $\pi \left(G\right)$ and two distinct vertices $p$ and $q$ are joined by an edge if and only if $G$ has an element of order $pq$. A group $G$ is called $k$-recognizable by prime graph if there exist exactly $k$ nonisomorphic groups $H$ satisfying the condition $\Gamma \left(G\right)=\Gamma \left(H\right)$. A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that $\mathrm{PGL}(2,p^{\alpha })$ is recognizable, if $p$ is an odd prime and $\alpha >1$ is odd. But for even $\alpha$, only...

Let $f:A\to B$ and $g:A\to C$ be two ring homomorphisms and let $K$ and $K^{\text{'}}$ be two ideals of $B$ and $C$, respectively, such that $f^{-1}\left(K\right)=g^{-1}\left(K^{\text{'}}\right)$. We investigate unipotent, symmetric and reversible properties of the bi-amalgamation ring $A{\bowtie }^{f,g}(K,K^{\text{'}})$ of $A$ with $(B,C)$ along $(K,K^{\text{'}})$ with respect to $(f,g)$.

Let $R$ be a noncommutative prime ring of characteristic different from 2, with its two-sided Martindale quotient ring $Q$, $C$ the extended centroid of $R$ and $a\in R$. Suppose that $\delta$ is a nonzero $\sigma$-derivation of $R$ such that $a{[\delta \left(x^n\right),x^n]}_k=0$ for all $x\in R$, where $\sigma$ is an automorphism of $R$, $n$ and $k$ are fixed positive integers. Then $a=0$.

Let $K_{s,t}$ be the complete bipartite graph with partite sets $\{x_1,...,x_s\}$ and $\{y_1,...,y_t\}$. A split bipartite-graph on $(s+s^{\text{'}})+(t+t^{\text{'}})$ vertices, denoted by ${\mathrm{SB}}_{s+s^{\text{'}},t+t^{\text{'}}}$, is the graph obtained from $K_{s,t}$ by adding $s^{\text{'}}+t^{\text{'}}$ new vertices $x_{s+1},...,x_{s+s^{\text{'}}}$, $y_{t+1},...,y_{t+t^{\text{'}}}$ such that each of $x_{s+1},...,x_{s+s^{\text{'}}}$ is adjacent to each of $y_1,...,y_t$ and each of $y_{t+1},...,y_{t+t^{\text{'}}}$ is adjacent to each of $x_1,...,x_s$. Let $A$ and $B$ be nonincreasing lists of nonnegative integers, having lengths $m$ and $n$, respectively. The pair $(A;B)$ is potentially ${\mathrm{SB}}_{s+s^{\text{'}},t+t^{\text{'}}}$-bigraphic if there is a simple bipartite graph containing ${\mathrm{SB}}_{s+s^{\text{'}},t+t^{\text{'}}}$ (with $s+s^{\text{'}}$ vertices $x_1,...,x_{s+s^{\text{'}}}$ in the part of size $m$...

Let $f:A\to B$ and $g:A\to C$ be two ring homomorphisms and let $J$ and $J^{\text{'}}$ be ideals of $B$ and $C$, respectively, such that $f^{-1}\left(J\right)=g^{-1}\left(J^{\text{'}}\right)$. In this paper, we investigate the transfer of the notions of Gaussian and Prüfer rings to the bi-amalgamation of $A$ with $(B,C)$ along $(J,J^{\text{'}})$ with respect to $(f,g)$ (denoted by $A{\bowtie }^{f,g}(J,J^{\text{'}})),$ introduced and studied by S. Kabbaj, K. Louartiti and M. Tamekkante in 2013. Our results recover well known results on amalgamations in C. A. Finocchiaro (2014) and generate new original examples of rings possessing these properties. ...

Let $r(G_1,G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V\left(S(n_1,n_2)\right)=\{v_0,v_1,...,v_{n_1},w_0$, $w_1,...,w_{n_2}\}$ and $E\left(S(n_1,n_2)\right)=\{v_0{v}_1,...,v_0{v}_{n_1},v_0{w}_0,w_0{w}_1,...,w_0{w}_{n_2}\}$. We determine $r(K_{1,m-1},$ $S(n_1,n_2))$ under certain conditions. For $n\ge 6$ let $T_n^3=S(n-5,3)$, $T_n^{\text{'}\text{'}}=(V,E_2)$ and $T_n^{\text{'}\text{'}\text{'}}=(V,E_3)$, where $V=\{v_0,v_1,...,v_{n-1}\}$, $E_2=\{v_0{v}_1,...,v_0{v}_{n-4},v_1{v}_{n-3}$, $v_1{v}_{n-2},v_2{v}_{n-1}\}$ and $E_3=\{v_0{v}_1,...,v_0{v}_{n-4},v_1{v}_{n-3},$ $v_2{v}_{n-2},v_3{v}_{n-1}\}$. We also obtain explicit formulas for $r(K_{1,m-1},T_n)$, $r(T_m^{\text{'}},T_n)$ $(n\ge m+3)$, $r(T_n,T_n)$, $r(T_n^{\text{'}},T_n)$ and $r(P_n,T_n)$, where $T_n\in \{T_n^{\text{'}\text{'}},T_n^{\text{'}\text{'}\text{'}},T_n^3\}$, $P_n$ is the path on $n$ vertices and $T_n^{\text{'}}$ is the unique tree with $n$ vertices and maximal degree $n-2$.

Let $R$ be a prime ring of characteristic different from 2, $Q_r$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$, $G$ are generalized skew derivations of $R$ with the same associated automorphism $\alpha$, and $p(x_1,...,x_n)$ is a non-central polynomial over $C$ such that $$\left[F\right(x),\alpha (y\left)\right]=G\left(\right[x,y\left]\right)$$ for all $x,y\in \{p(r_1,...,r_n):r_1,...,r_n\in R\}$. Then there exists $\lambda \in C$ such that $F\left(x\right)=G\left(x\right)=\lambda \alpha \left(x\right)$ for all $x\in R$.

Given a graph $G$, let $f\left(G\right)$ denote the maximum number of edges in a bipartite subgraph of $G$. Given a fixed graph $H$ and a positive integer $m$, let $f(m,H)$ denote the minimum possible cardinality of $f\left(G\right)$, as $G$ ranges over all graphs on $m$ edges that contain no copy of $H$. In this paper we prove that $f(m,{\theta }_{k,s})\ge \frac{1}{2}m+\Omega \left(m^{(2k+1)/(2k+2)}\right)$, which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write $K_k^{\text{'}}$ and $K_{t,s}^{\text{'}}$ for the subdivisions of $K_k$ and $K_{t,s}$. We show that $f(m,K_k^{\text{'}})\ge \frac{1}{2}m+\Omega \left(m^{(5k-8)/(6k-10)}\right)$ and $f(m,K_{t,s}^{\text{'}})\ge \frac{1}{2}m+\Omega \left(m^{(5t-1)/(6t-2)}\right)$, improving a result of Q. Zeng, J. Hou. We also give lower bounds on...

Let $G$ be a group and $\omega \left(G\right)$ be the set of element orders of $G$. Let $k\in \omega \left(G\right)$ and $m_k\left(G\right)$ be the number of elements of order $k$ in $G$. Let nse$\left(G\right)=\{m_k\left(G\right):k\in \omega \left(G\right)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$\left(G\right)=$ nse$\left(S_r\right)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

Let $R$ be a prime ring with center $Z\left(R\right)$ and $I$ a nonzero right ideal of $R$. Suppose that $R$ admits a generalized reverse derivation $(F,d)$ such that $d\left(Z\right(R\left)\right)\ne 0$. In the present paper, we shall prove that if one of the following conditions holds: (i) $F\left(xy\right)\pm xy\in Z\left(R\right)$, (ii) $F\left(\right[x,y\left]\right)\pm \left[F\right(x),y]\in Z\left(R\right)$, (iii) $F\left(\right[x,y\left]\right)\pm \left[F\right(x),F(y\left)\right]\in Z\left(R\right)$, (iv) $F(x\circ y)\pm F\left(x\right)\circ F\left(y\right)\in Z\left(R\right)$, (v) $\left[F\right(x),y]\pm [x,F(y\left)\right]\in Z\left(R\right)$, (vi) $F\left(x\right)\circ y\pm x\circ F\left(y\right)\in Z\left(R\right)$ for all $x,y\in I$, then $R$ is commutative.

Given an integer $n\ge 3$, let $u_1,...,u_n$ be pairwise coprime integers $\ge 2$, $𝒟$ a family of nonempty proper subsets of $\{1,...,n\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_i-\epsilon \left(I\right)$ for some $I\in 𝒟$, but it does not divide $u_1\cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\epsilon$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if ${\epsilon }_0\in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

Let $K$ be a number field defined by an irreducible polynomial $F\left(X\right)\in \mathbb{Z}\left[X\right]$ and ${\mathbb{Z}}_K$ its ring of integers. For every prime integer $p$, we give sufficient and necessary conditions on $F\left(X\right)$ that guarantee the existence of exactly $r$ prime ideals of ${\mathbb{Z}}_K$ lying above $p$, where $\overline{F}\left(X\right)$ factors into powers of $r$ monic irreducible polynomials in ${𝔽}_p\left[X\right]$. The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly $r$ prime ideals of ${\mathbb{Z}}_K$ lying above $p$....