is a module over the ring of all modular forms with respect to the group Γ2[4, 8]. We are interested in its structure. By Igusa, the ring of modular forms is generated by the ten classical theta constants θ[m]. The module M contains a submodule N which is generated by 45 Cohen-Rankin brackets {θ[m], θ[n]}. We determine defining relations for this submodule and compute its Hilbert function (Theo...

In this paper, we introduce the concept of a quasi-radical semi prime submodule. Throughout work, assume that is commutative ring with identity and left unitary R- module. A proper submodule called (for short Q-rad-semiprime), if for , ,and then . Where intersection all submodules

Let $G$ be a group with identity $e$. $R$ $G$-graded commutative ring and $M$ graded $R$-module. In this paper, we introduce the concept of $WAG2$-absorbing submodule. A number results concerning these classes submodules their homogeneous components are given.
 $N=\bigoplus _{h\in G}N_{h}$ submodule $h\in G.$ We say that $N_{h}$ is $h$-$WAG2$-absorbing $R_{e}$-module $M_{h}$ if $N_{h}\neq ...

Given an n-bit integer N , output YES if n is prime and NO otherwise. This is one of the most basic questions about numbers, with the following history. • By definition Prime ∈ coNP, because the prime decomposition is a short certificate for a number that is not prime. • [Pratt'75] showed that Prime ∈ NP. The Pratt certificate of a number N being prime, is by looking at all prime factor q of N ...

Let M be a graded leftA-module and M∗ the associate complex of M. Then : If is noetherian (resp. artinian) then strongly hopfian cohopfian); cohopfian), leftA-module, M, N submodule N∗ fully invariant subcomplex M∗. M∗/N∗ hopfian, hopfian. if all cohopfian, cohopfian.

The Hardy space on the unit ball in C provides examples of a quasi-free, finite rank Hilbert module which contains a pure submodule isometrically isomorphic to the module itself. For n = 1 the submodule has finite codimension. In this note we show that this phenomenon can only occur for modules over domains in C and for finitely-connected domains only for Hardy-like spaces, the bundle shifts. M...

A nonempty subset A of {1, 2, . . . , n} is said to be relatively prime if gcd(A) = 1. Nathanson [4] defined f(n) to be the number of relatively prime subsets of {1, 2, . . . , n} and, for k ≥ 1, fk(n) to be the number of relatively prime subsets of {1, 2, . . . , n} of cardinality k. Nathanson [4] defined Φ(n) to be the number of nonempty subsets A of the set {1, 2, . . . , n} such that gcd(A)...

Let f(m,n) denote the number of relatively prime subsets of {m + 1,m + 2, . . . , n}, and let Φ(m,n) denote the number of subsets A of {m+1,m+2, . . . , n} such that gcd(A) is relatively prime to n. Let fk(m,n) and Φk(m,n) be the analogous counting functions restricted to sets of cardinality k. Simple explicit formulas and asymptotic estimates are obtained for these four functions. A nonempty s...