Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a containing line in each direction. We show that, when $N$ any square-free integer, the size of smallest $(\mathbb{Z}/N\mathbb{Z})^n$ at least $C_{n,\epsilon} N^{n - \epsilon}$ for $\epsilon$ -- resolving special case conjecture Hickman and Wright. Previously, such bounds were only known prime $N$. also that general can be reduced to lower...

Let K be a number field, with degree n and ring of integers OK . When OK = Z[α] for some α ∈ OK , the set {1, α, . . . , αn−1} is a Z-basis of OK . We call such a basis a power basis. When K is a quadratic field or a cyclotomic field, OK admits a power basis, and the use of these two fields as examples in algebraic number theory can lead to the impression that rings of integers should always ha...

For various positive integers k, the sums of kth powers of the first n positive integers, Sk(n) := 1 k+2k+ · · ·+nk, are some of the most popular sums in all of mathematics. In this note we prove a congruence modulo n3 involving two consecutive sums S2k(n) and S2k+1(n). This congruence allows us to establish an equivalent formulation of Giuga’s conjecture. Moreover, if k is even and n ≥ 5 is a ...

We will start with introducing congruences and investigating modular arithmetic: the set Z/nZ of “integers modulo n” forms a ring. This ring is a field if and only if n is a prime number. A study of the multiplicative structure leads to Fermat’s Little Theorem (for prime n) and to the Euler phi function and Euler’s generalization of Fermat’s Theorem. Another basic tool is the Chinese Remainder ...

γ = c0 + c1p+ c2p + · · · = (. . . c3c2c1c0)p, with ci ∈ Z, 0 ≤ ci ≤ p− 1, called the digits of γ. This ring has a topology given by a restriction of the product topology—we will see this below. The ring Zp can be viewed as Z/pZ for an ‘infinitely high’ power n. This is a useful idea, for example, in the study of Diophantine equations: if such an equation has a solution in the integers, then it...

The coprime graph of a group G is Γ_G with its set vertices and any two distinct are adjacent if only their order relatively prime. Let p be prime number, then G_p denotes the multiplicative 2×2 upper unitriangular matrices over ring integers modulo p. purposes this research to study Γ_(G_p ) find first second Zagreb eccentricity indices for p≥3. results as follows. First index )isE_1 (Γ_(G_p )...

We prove that if a subset of the d-dimensional vector space over the ring of integers modulo pr is large enough, then the number of k-tuples of mutually orthogonal vectors in this set is close to its expected value.