the submodules with the property of the title ( a submodule $n$ of an $r$-module $m$ is called strongly dense in $m$, denoted by $nleq_{sd}m$, if for any index set $i$, $prod _{i}nleq_{d}prod _{i}m$) are introduced and fully investigated. it is shown that for each submodule $n$ of $m$ there exists the smallest subset $d'subseteq m$ such that $n+d'$ is a strongly dense submodule of $m$...

We prove that $L(SL_2(\textbf{k}))$ is a maximal Haagerup von Neumann subalgebra in $L(\textbf{k}^2\rtimes SL_2(\textbf{k}))$ for $\textbf{k}=\mathbb{Q}$. Then we show how to modify the proof handle $\textbf{k}=\mathbb{Z}$. The key step complete description of all intermediate subalgebras between and $L^{\infty}(Y)\rtimes SL_2(\textbf{k})$, where $SL_2(\textbf{k})\curvearrowright Y$ denotes quo...

Let $M$ be a module over a commutative ring $R$ and let $N$ be a proper submodule of $M$. The total graph of $M$ over $R$ with respect to $N$, denoted by $T(Gamma_{N}(M))$, have been introduced and studied in [2]. In this paper, A generalization of the total graph $T(Gamma_{N}(M))$, denoted by $T(Gamma_{N,I}(M))$ is presented, where $I$ is an ideal of $R$. It is the graph with all elements of $...

The human monocyte (M phi subset rosetting with anti RH-coated human erythrocytes via high-affinity, 72 kD receptors (FcRI+), contains the PGE2-producing immunosuppressive subpopulation, while the non-rosetting M phi subset (FcRI-) is the major plasminogen activator-producing and antigen-presenting M phi. This study gives additional evidence for the functional disparity of the FcRI- and FcRI+ M...

Let R be a Noetherian ring, F := Rr and M ⊆ F a submodule of rank r. Let A∗(M) denote the stable value of Ass(Fn/Mn), for n large, where Fn is the nth symmetric power of Fn and Mn is the image of the nth symmetric power of M in Fn. We provide a number of characterizations for a prime ideal to belong to A∗(M). We also show that A∗(M) ⊆ A∗(M), where A∗(M) denotes the stable value of Ass(Fn/Mn).

Abstract The purposes of this paper are to introduce generalizations quasi-prime ideals the context $$\phi $$ ? -quasi-prime ideals. Let : {\mathcal {I}}(S) \rightarrow \cup \left\{ \emptyset \right\} : I ( S ) </m...

A graph G(V,E) with vertex set V is said to have a prime labeling if there exist a bijection f : V (G) → {1, 2, . . . , |V |} such that for each edge xy ∈ E(G), f(x) and f(y) are relatively prime. It becomes an interesting problem to investigate the maximum possible edges that can be constructed in a graph with n vertices having prime labeling property. We interpret this problem in number theor...

Let $R\ $be a commutative ring with $1\neq0$ and $M$ be an $R$-module. Suppose that $S\subseteq R\ $is multiplicatively closed set of $R.\ $Recently Sevim et al. in \cite{SenArTeKo} introduced the notion $S$-prime submodule which is generalization prime used them to characterize certain classes rings/modules such as submodules, simple modules, torsion free modules,\ $S$-Noetherian modules etc. ...

The work in this paper is inspired and motivated by some work of Nathanson. We count the number of relatively prime subsets and the number of relatively prime subsets having some fixed cardinality that are in {m,m+1, . . . , n}. We also count the number of nonempty subsets of {m,m+1, . . . , n} whose gcd is relatively prime to n and the number of nonempty subsets {m,m+1, . . . , n} having some ...

The immunopathogenesis of autoimmune insulin-dependent diabetes in NOD mice entails defects in the development of macrophages (M phi s) from hematopoietic precursors. The present study analyzes the cellular and molecular basis underlying our previous finding that the Mø growth factor colony-stimulating factor 1 (CSF-1) promotes a reduced level of promonocyte proliferation and M phi development ...