In this paper we characterize the radical of an arbitrary‎ ‎submodule $N$ of a finitely generated free module $F$ over a‎ ‎commutatitve ring $R$ with identity‎. ‎Also we study submodules of‎ ‎$F$ which satisfy the radical formula‎. ‎Finally we derive‎ ‎necessary and sufficient conditions for $R$ to be a‎ ‎Pr$ddot{mbox{u}}$fer domain‎, ‎in terms of the radical of a‎ ‎cyclic submodule in $Rbigopl...

In this paper we introduce the concept of weakly prime ternary subsemimodules of a ternary semimodule over a ternary semiring and obtain some characterizations of weakly prime ternary subsemimodules. We prove that if $N$ is a weakly prime subtractive ternary subsemimodule of a ternary $R$-semimodule $M$, then either $N$ is a prime ternary subsemimodule or $(N : M)(N : M)N = 0$. If $N$ is a $Q$-...

In this paper, for any prime $p$, we propose the notion of a $p$-transitive association scheme. This aims to generalize fact that regular module group algebra finite has unique trivial submodule case modules modular adjacency algebras. We completely determine quasi-thin schemes and with thin residue by their structure theory properties.

Let φ denote Euler’s phi function. For a fixed odd prime q we give an asymptotic series expansion in the sense of Poincaré for the number Eq(x) of n ≤ x such that q ∤ φ(n). Thereby we improve on a recent theorem by B.K. Spearman and K.S. Williams [Ark. Mat. 44 (2006), 166–181]. Furthermore we resolve, under the Generalized Riemann Hypothesis, which of two approximations to Eq(x) is asymptotical...

let $r$ be a commutative ring with identity and $m$ be a unitary$r$-module. the primary-like spectrum $spec_l(m)$ is thecollection of all primary-like submodules $q$ such that $m/q$ is aprimeful $r$-module. here, $m$ is defined to be rsp if $rad(q)$ isa prime submodule for all $qin spec_l(m)$. this class containsthe family of multiplication modules properly. the purpose of thispaper is to intro...

In this paper, all rings are associative with identity and all modules are unital left modules unless otherwise specified. Let R be a ring and M a module. N ≤M will mean N is a submodule of M. A submodule E of M is called essential in M (notation E ≤e M) if E∩A = 0 for any nonzero submodule A of M. Dually, a submodule S of M is called small in M (notation S M) if M = S+T for any proper submodul...

Let $p$ be a prime, let $K$ discretely valued extension of $\mathbb{Q}_p$, and $A_{K}$ an abelian $K$-variety with semistable reduction. Extending work by Kim Marshall from the case where $p>2$ $K/\mathbb{Q}_p$ is unramified, we prove $l=p$ complement Galois cohomological formula Grothendieck for $l$-primary part N\'eron component group $A_{K}$. Our proof involves constructing, each $m\in \math...

The flavivirus bovine viral diarrhoea (BVD) virus exists in two biotypes, cytopathic (cp) and non-cytopathic (ncp), defined by their effect on cultured cells. Cp BVD virus-infected cells undergo apoptosis and may promote apoptosis in uninfected cells by an indirect mechanism. Macrophages (Mφ) infected with cp, but not ncp, BVD virus release a factor(s) in the supernatant capable of priming ...

Motivated by a question of Venkataramana, we consider the greatest common divisor $\phi(f(n))$ where $f$ is primitive polynomial with integer coefficients, and $n$ ranges over all natural numbers. Assuming Schinzel's hypothesis, establish that this gcd may be bounded just in terms degree $f$. Unconditionally such bound for quadratic polynomials, as well polynomials split completely into linear ...

Abstract Let $${\mathcal {T}}_\infty (F)$$ T∞(F) be the algebra of all $${\mathbb {N}}\times {\mathbb {N}}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">N</mml:...