ardy and Littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. They believed that the number $mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically given by begin{equation*} mathcal{R}(n) sim frac{sqrt{n}}{log n}prod_{p=3}^{infty}left(1-frac{1}{p-1}left(frac{n}{p}right)right), end{equation*} where $p$ is a prime, $m$ is a...

In this paper, persents the definitions of strongly prime ideal, strongly prime N-subgroup, Pseudo-valuation near ring and Pseudo-valuation N-group. Some of their properties have also been proven by theorems. Then it is shown that, if N be near ring with quotient near-field K and P be a strongly prime ideal of near ring N, then is a strongly prime ideal of ‎‎, for any multiplication subset S of...

in this paper, we give a generalization of the integral dependence from rings to modules. we study the stability of the integral closure with respect to various module theoretic constructions. moreover, we introduce the notion of integral extension of a module and prove the lying over, going up and going down theorems for modules.

A finitely generated $R$-module is said to be a module of type ($F_r$) if its $(r-1)$-th Fitting ideal is the zero ideal and its $r$-th Fitting ideal is a regular ideal. Let $R$ be a commutative ring and $N$ be a submodule of  $R^n$ which is generated by columns of  a matrix $A=(a_{ij})$ with $a_{ij}in R$ for all $1leq ileq n$, $jin Lambda$, where $Lambda $ is a (possibly infinite) index set.  ...

In the present paper, by considering the notion of MV-modules which is the structure that naturally correspond to lu-modules over lu-rings, we prove some results on prime A-ideals and state some conditions to obtain a prime A-ideal in MV-modules. Also, we state some conditions that an A-ideal is not prime and investigate conditions that $Ksubseteq bigcup _{i=1}^{n}K_{i}$ implies $Ksubseteq K_{j...

let $r$ be a commutative ring with identity and $m$ be an$r$-module. let $fspec(m)$ denotes the collection of all prime fuzzysubmodules of $m$. in this regards some basic properties of zariskitopology on $fspec(m)$ are investigated. in particular, we provesome equivalent conditions for irreducible subsets of thistopological space and it is shown under certain conditions$fspec(m)$ is a $t_0-$spa...

let $r$ be a ring‎, ‎and let $n‎, ‎d$ be non-negative integers‎. ‎a right $r$-module $m$ is called $(n‎, ‎d)$-projective if $ext^{d+1}_r(m‎, ‎a)=0$ for every $n$-copresented right $r$-module $a$‎. ‎$r$ is called right $n$-cocoherent if every $n$-copresented right $r$-module is $(n+1)$-coprese-nted‎, ‎it is called a right co-$(n,d)$-ring if every right $r$-module is $(n‎, ‎d)$-projective‎. ‎$r$ ...

 Let $R$ be a ring‎, ‎and let $n‎, ‎d$ be non-negative integers‎. ‎A right $R$-module $M$ is called $(n‎, ‎d)$-projective if $Ext^{d+1}_R(M‎, ‎A)=0$ for every $n$-copresented right $R$-module $A$‎. ‎$R$ is called right $n$-cocoherent if every $n$-copresented right $R$-module is $(n+1)$-coprese-nted‎, ‎it is called a right co-$(n,d)$-ring if every right $R$-module is $(n‎, ‎d)$-projective‎. ‎$R$...

we prove the generalized hyers--ulam stability  of n--th order linear differential equation of the form $y^{(n)}+p_{1}(x)y^{(n-1)}+ cdots+p_{n-1}(x)y^{prime}+p_{n}(x)y=f(x)$, with condition that there exists a non--zero solution of corresponding homogeneous equation. our main results extend and improve the corresponding results obtained by many authors.