In 1988 P. Erdös asked if the prime divisors of x−1 for all n = 1, 2, . . . determine the given integer x; the problem was affirmatively answered by Corrales-Rodorigáñez and R. Schoof [2] in 1997 together with its elliptic version. Analogously, K. Yamanoi [14] proved in 2004 that the support of the pull-backed divisor fD of an ample divisor on an abelian variety A by an algebraically non-degene...

This paper presents a selection of lexicographic codes over a prime field Fp , codes constructed so as to be linear and to have a prescribed minimum distance and divisor. The development of the code when the minimum distance is 3p and the divisor is p leads to questions involving the distribution of quadratic residues and non-residues modulo p. In the course of events, the Hall planeH(p) emerge...

Let X be a smooth, complex Fano 4-fold, and ρX its Picard number. If contains prime divisor D with ρX−ρD>2, then either is product of del Pezzo surfaces, or ρX=5,6. In this setting, we completely classify the case where ρX=5; there are 6 families, among which one new. We also deduce classification 4-folds ρX≥5 an elementary divisorial contraction sending to curve.

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular we derive that in any local ring R of mixed...

Many authors studied the graph theory in connection with commutative semigroups and commutative and noncommutative rings as we can refer to references. For example, Beck 1 associated to any commutative ring R its zero-divisor graph G R whose vertices are the zero-divisors of R including 0 , with two vertices a, b joined by an edge in case ab 0. Also, DeMeyer et al. 2 defined the zero-divisor gr...

Let X be an irreducible affine algebraic variety that is spherical with respect to action of a connected reductive group G. In this paper, we provide sufficient conditions, formulated in terms weight combinatorics, for the existence one-parameter additive actions on $$X$$ normalized by Borel subgroup $$B \subset G$$ . As application, prove every G-stable prime divisor can open G-orbit means sui...

Denote by G a finite group and let ? ( ) denote the sum of element orders in . In 2009, H. Amiri, S.M. Jafarian Amiri I.M. Isaacs proved that if | = n is non-cyclic, then < C , where denotes cyclic order 2018 we non-cyclic ? 7 11 equality holds 4 k with 2 1 × this paper only are as indicated above. Moreover following generalization result: Theorem Let be q being least prime divisor Then ? + 5 N...

In this research, modelling of the leaves with help divisor functions are worked on. Given a positive integer k (1 ≤ 100), we investigate solutions equation σ(n) = σ(n + 2k) odd square-free n. Further, for l and prime q, there no results σ2i(n) σ2i(q). As an application, pose basic structure model real-time virtual ecosystem construction derived from shifted functions. Also, elliptic, flabellat...

Let φ(n) denote Euler’s totient function, i.e., the number of positive integers < n and prime to n. We study pairs of positive integers (a0, a1) with a0 ≤ a1 such that φ(a0) = φ(a1) = (a0 + a1)/k for some integer k ≥ 1. We call these numbers φ–amicable pairs with multiplier k, analogously to Carmichael’s multiply amicable pairs for the σ–function (which sums all the divisors of n). We have comp...

It is shown that a commutative Bézout ring R with compact minimal prime spectrum is an elementary divisor ring if and only if so is R/L for each minimal prime ideal L. This result is obtained by using the quotient space pSpec R of the prime spectrum of the ring R modulo the equivalence generated by the inclusion. When every prime ideal contains only one minimal prime, for instance if R is arith...