Narrow Results

We describe primitive and prime ideals in the C*-algebra C*(E) of a graph E satisfying Condition (K), together with the topologies on each of these spaces. In particular, we find that primitive ideals correspond to the set of maximal tails disjoint union the set of finite-return vertices, and that prime ideals correspond to the set of clusters of maximal tails disjoint union the set of finite-return vertices.

We prove a folklore theorem of Thurston, which provides necessary and sufficient conditions for primality of a certain class of theta-curves. Namely, a theta-curve in the 3-sphere with an unknotted constituent knot $\kappa$ is prime, if and only if lifting the third arc of the theta-curve to the double branched cover over $\kappa$ produces a prime knot. We apply this result to Kinoshita’s theta-curve.

In this paper we study prime preradicals, irreducible preradicals, ∧-prime preradicals, prime submodules and diuniform modules. We study some relations between these concepts, using the lattice structure of preradicals developed in previous papers. In particular, we give a characterization of prime preradicals using an operator named the relative annihilator. We also characterize prime submodules by means of prime preradicals. We give some characterizations of rings that have certain conditions on prime radicals and on irreducible preradicals, such as left local left V-rings, as well as 1-spr rings, which we introduce.

In this paper we characterize *-prime group rings. We prove that the group ring RG of the group G over the ring R is *-prime if and only if R is *-prime and Λ⁺(G) = (1). In the process we obtain more examples of group rings which are *-prime but not strongly prime.

Many radicals are determined by a property that subsets of a ring may satisfy; for example, the subsets of a ring singled out could be m-systems leading to the prime radical. Here this process is formalized. In addition, the conditions that a property of the subsets must satisfy to ensure that the corresponding radical fulfills a variety of criteria are described.

We show that associative systems with a sufficiently good module structure imbed in a primitive system with simple primitive heart, spanned by the original system and the heart, so extending the results of J. Pure Appl. Algebra 181 (2003) 131–139 to systems over more general rings of scalars. We also study associative systems with involution.

We study a certain type of prime Noetherian idealiser ring R of injective dimension 1, and prove for instance that the idempotent ideals of R are projective and that every non-zero projective ideal of R is uniquely of the form UE for some invertible ideal U and idempotent ideal E of R. Formulae are given for the number of idempotent ideals of R and the number of orders which contain R.

The flow (or lack thereof) of several kinds of primeness between a zero-symmetric near-ring R and its group near-ring R[G] for certain groups G is discussed. In certain cases, results are contrasted against what happens in the matrix near-ring situation.

We show that the set of the numbers that are the sum of a prime and a Fibonacci number has positive lower asymptotic density.

Let $k$ be an integer with $k\ge 4$ and $\eta$ be any real number. Suppose that ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$ are nonzero real numbers, not all of the same sign and ${\lambda }_1/{\lambda }_2$ is irrational. It is proved that the inequality

It is proved that if ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_5$ are nonzero real numbers, not all of the same sign and ${\lambda }_1/{\lambda }_2$ is irrational, then for given real numbers $\eta$ and $\sigma$, $0<\sigma <\frac{5}{252}$, the inequality

Let $k$ be an integer with $k\ge 3$, and $\eta$ be any real number. Suppose that ${\lambda }_0,{\lambda }_1,\dots ,{\lambda }_s$ are nonzero real numbers, not all the same sign and ${\lambda }_0/{\lambda }_1$ is irrational. It is proved that the inequality $|{\lambda }_0{p}_0+{\lambda }_1{p}_1^k+{\lambda }_2{p}_2^k+\cdots +{\lambda }_s{p}_s^k+\eta |<{({max}_{1\le j\le s}{p}_j)}^{-\sigma (k)+𝜀}$ has infinitely many solutions in primes $p_0,\dots ,p_s$, where $\sigma (3)=\frac{1}{14}$, and $\sigma (k)={(3\cdot 2^{k-1})}^{-1}$ for $k\ge 4$. This generalizes earlier results. As application, we get that the integer parts of ${\lambda }_1{p}_1^k+{\lambda }_2{p}_2^k+\cdots +{\lambda }_s{p}_s^k$ are prime infinitely often for primes $p_1,p_2,\dots ,p_s$.

Let $1<c<\frac{43}{36}$. In this paper, it is proved that for every sufficiently large real number $N$, the Diophantine inequality

Let $[𝜃]$ denote the integral part of the real number $𝜃$. In this paper, it is proved that for $2<c<\frac{408}{197}$, the Diophantine equation $[p_1^c]+[p_2^c]+[p_3^c]+[p_4^c]+[p_5^c]=N$ is solvable in prime variables $p_1,\dots ,p_5$ for sufficiently large integer $N$.

We prove that there exists positive even integer $k$ such that $\varphi (n)=\varphi (n+k)$ holds for infinitely many $n$. We also prove various estimates on number of solutions to $\varphi (p-1)=\varphi (q-1)$ for distinct primes $p$ and $q$.

In this paper, we show that the lower density of integers representable as the sum of a prime and a Fibonacci number is at least 0.0254905.

In this paper, we show that there are infinitely many primes p such that $p+2$ has at most two prime factors and $p+6$ has at most 11 prime factors. This result constitutes an improvement on that of Cai.

In this paper we define prime, semiprime and irreducible ideals in ternary semigroups. We also define semisimple ternary semigroups and prove that a ternary semigroup is semisimple if and only if each of its ideals is semiprime.

In a graph $G=(V,E)$, a module is a vertex subset $M$ of $V$ such that every vertex outside $M$ is adjacent to all or none of $M$. For example, $\varnothing$, $\{x\}$$(x\in V)$ and $V$ are modules of $G$, called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable. A vertex $x$ of a prime graph $G$ is critical if $G-x$ is decomposable. Moreover, a prime graph with $k$ noncritical vertices is called $(-k)$-critical graph. A prime graph $G$ is $k$-minimal if there is some $k$-vertex set $X$ of vertices such that there is no proper induced subgraph of $G$ containing $X$ is prime. From this perspective, Boudabbous proposes to find the $(-k)$-critical graphs and $k$-minimal graphs for some integer $k$ even in a particular case of graphs. This research paper attempts to answer Boudabbous’s question. First, we describe the $(-k)$-critical tree. As a corollary, we determine the number of nonisomorphic $(-k)$-critical tree with $n$ vertices where $k\in \{1,2,\lfloor \frac{n}{2}\rfloor \}$. Second, we provide a complete characterization of the $k$-minimal tree. As a corollary, we determine the number of nonisomorphic $k$-minimal tree with $n$ vertices where $k\le 3$.

Let I be an ideal of a nearring N. We introduce the notions of equiprime graph of N denoted by EQ_I (N) and c-prime graph of N denoted by C_I (N). We relate EQ_I (N), C_I (N) and the graph of a nearring with respect to an ideal, G_I (N). We prove that diam(EQ_I (N\I)) ≤ 3 and diam(C_I (N\I)) ≤ 3 and deduce that the prime graphs are edge partitionable. It is well-known that the homomorphic image of a prime ideal need not be a prime ideal in general. We study graph homomorphisms and obtain conditions under which the primeness property of an ideal is preserved under nearring homomorphisms.