‎let $g$ be a group with identity $e.$ let $r$ be a $g$-graded‎ ‎commutative ring and $m$ a graded $r$-module‎. ‎in this paper‎, ‎we‎ ‎introduce several results concerning graded classical prime‎ ‎submodules‎. ‎for example‎, ‎we give a characterization of graded‎ ‎classical prime submodules‎. ‎also‎, ‎the relations between graded‎ ‎classical prime and graded prime submodules of $m$ are studied‎.‎

‎there are a few finite groups that are determined up to isomorphism solely by their order, such as $mathbb{z}_{2}$ or $mathbb{z}_{15}$. still other finite groups are determined by their order together with other data, such as the number of elements of each order, the structure of the prime graph, the number of order components, the number of sylow $p$-subgroups for each prime $p$, etc. in this...

The concept of gyrogroups, with a weaker algebraic structure without associative law, was introduced under the background c -ball relativistically admissible velocities Einstein velocity addition. A topological gyrogroup is just endowed compatible topology such that multiplication jointly continuous and inverse continuous. This generalizes groups. In this paper, we are going to establish for lo...

Let Γ = Cay(G,S) be a Cayley digraph on a group G and let A = Aut(Γ). The Cayley index of Γ is |A : G|. It has previously been shown that, if p is a prime, G is a cyclic p-group and A contains a noncyclic regular subgroup, then the Cayley index of Γ is superexponential in p. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contras...

The paper uses a new approach to investigate prime submodules and minimal prime submodules of certain modules such as Artinian and torsion modules. In particular, we introduce a concrete formula for the radical of submodules of Artinian modules.

In 1976, S. Winograd [20] presented a new DFT algorithm which had signi cantly fewer multiplications than the Cooley-Tukey FFT which had been published eleven years earlier. This new Winograd Fourier Transform Algorithm (WFTA) is based on the typeone index map from Multidimensional Index Mapping with each of the relatively prime length short DFT's calculated by very e cient special algorithms. ...

‎let $g$ be a graph and $chi^{prime}_{aa}(g)$ denotes the minimum number of colors required for an‎ ‎acyclic edge coloring of $g$ in which no two adjacent vertices are incident to edges colored with the same set of colors‎. ‎we prove a general bound for $chi^{prime}_{aa}(gsquare h)$ for any two graphs $g$ and $h$‎. ‎we also determine‎ ‎exact value of this parameter for the cartesian product of ...

Let Zn denote the cyclic group of n elements. For every x ∈ Zn, we denote by |x|n the smallest positive integer in the residual class x. Let S = (a1, . . . , ak) be a sequence of elements in Zn. We say that S has index n if there is an integer m co-prime to n such that Pk i=1 |mai|n = n. In this paper, we prove that a sequence of n elements in Zn with repetition at least n 2 contains a subseque...

We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson-Lichtenbaum conjecture, we apply our results to show the vanishing of SK2(A) for a central simple algebra A of square-free index (prime to the characteristic). This pro...