f : N × R → R, K : N × N → R, K(n, i) = 0 for n < i, and b : N → R. We regard N× R as a metric subspace of the Euclidean plane R2. By a solution of (E) we mean a sequence x : N→ R satisfying (E) for all large n. We say that x is a full solution of (E) if (E) is satisfied for all n. Moreover, if p ∈ N and (E) is satisfied for all n ≥ p, then we say that x is a p-solution. For the sake of conveni...

We begin a systematic study of those finite semigroups that generate join irreducible members of the lattice of pseudovarieties of finite semigroups, which are important for the spectral theory of this lattice. Finite semigroups S that generate join irreducible pseudovari-eties are characterized as follows: whenever S divides a direct product A × B of finite semigroups, then S divides either A ...

Cohn called a ring $R$ is reversible if whenever $ab = 0,$ then $ba = 0$ for $a,bin R.$ The reversible property is an important role in noncommutative ring theory‎. ‎Recently‎, ‎Abdul-Jabbar et al‎. ‎studied the reversible ring property on nilpotent elements‎, ‎introducing‎ the concept of commutativity of nilpotent elements at zero (simply‎, ‎a CNZ ring)‎. ‎In this paper‎, ‎we extend the CNZ pr...

In Theorem 2, we shall show that the only integers having property P(n), where n is an odd positive integer, are -2, -1, 1, and 2. In Theorem 3, we shall determine the integers which have property P(n) , where n is an even positive integer. In particular, we shall show that: m has property P(4) iff m divides 240 = 2̂ 3 -5 m has property P(6) iff m divides 504 = 23 * 7 m has property P(8) iff m d...

Let $G$ be a group with identity $e$ and $R$ be a commutative $G$-graded ring with nonzero unity $1$. In this article, we introduce the concept of graded $r$-ideals. A proper graded ideal $P$ of a graded ring $R$ is said to be graded $r$-ideal if whenever $a, bin h(R)$ such that $abin P$ and $Ann(a)={0}$, then $bin P$. We study and investigate the behavior of graded $r$-ideals to introduce ...

Derived forms defined by M. Aschbacher in [1] are closely related to combinatorial polarization introduced by H. N. Ward in [6]. A binary linear code is said to be of (divisibility) level r, if r is the biggest integer such that 2r divides the weight of each codeword. In this paper, we study the relation between functions of combinatorial degreee r+1 and binary linear codes of level r. We assoc...

Let $G$ be a $p$-group of nilpotency class $k$ with finite exponent $exp(G)$ and let $m=lfloorlog_pk floor$. We show that $exp(M^{(c)}(G))$ divides $exp(G)p^{m(k-1)}$, for all $cgeq1$, where $M^{(c)}(G)$ denotes the c-nilpotent multiplier of $G$. This implies that $exp( M(G))$ divides $exp(G)$, for all finite $p$-groups of class at most $p-1$. Moreover, we show that our result is an improvement...

Let $R$ be a ring, $n$ an non-negative integer and $d$ positive or $\infty$. A right $R$-module $M$ is called \emph{$(n,d)^*$-projective} if ${\rm Ext}^1_R(M, C)=0$ for every $n$-copresented $C$ of injective dimension $\leq d$; ring \emph{right $(n,d)$-cocoherent} with $id(C)\leq d$ $(n+1)$-copresented; $(n,d)$-cosemihereditary} whenever $0\rightarrow C\rightarrow E\rightarrow A\rightarrow 0$ e...

Let $R$ be a commutative ring with identity and $N(R)$ and&#x0D; $J\left(R\right)$ denote the nilradical Jacobson radical&#x0D; of $R$, respectively. A proper ideal $I$ is called an&#x0D; n-ideal if for every $a,b\in R$, whenever $ab\in I$\ $a\notin&#x0D; N(R)$, then $b\in I$. In this paper, we introduce study&#x0D; J-ideals as new generalization n-ideals in rings.&#x0D; $I$\ $R$\ J-ideal $ab\i...

A Somos 4 sequence is a sequence (hn) of rational numbers defined by the quadratic recursion hm+2 hm−2 = λ1 hm+1 hm−1 + λ2 h 2 m for all m ∈ Z for some rational constants λ1, λ2. Elliptic divisibility sequences or EDSs are an important special case where λ1 = h 2 2, λ2 = −h1 h3, the hn are integers and hn divides hm whenever n divides m. Somos (4) is the particular Somos 4 sequence whose coeffi...