let r be a g-graded ring and m be a g-graded r-module. in this article, we introduce the concept of graded weakly classical prime submodules and give some properties of such submodules.

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...

We present a modular algorithm for computing GCDs of univariate polynomials with coefficients modulo a zero-dimensional triangular set. Our algorithm generalizes previous work for computing GCDs over algebraic number fields. The main difficulty is when a zero divisor is encountered modulo a prime number. We give two ways of handling this: Hensel lifting, and fault tolerant rational reconstructi...

Given a well-chosen additively homomorphic cryptosystem and a Σ protocol with a linear answer, Damg̊ard, Fazio, and Nicolosi proposed a non-interactive designated-verifier zero knowledge argument in the registered public key model that is sound under non-standard complexity-leveraging assumptions. In 2015, Chaidos and Groth showed how to achieve the weaker yet reasonable culpable soundness notio...

We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution of curves, the complexity is subexponential in genus 2, and O(N1/12) in genus 3. We give example...

We explicitly compute the homology groups with coefficients in a field of characteristic zero cocyclic subgroups or even Artin FC-type. also give some partial results case when are taken prime characteristic.

Duan and Winter studied the one-shot zero-error classical capacity of a quantum channel assisted by quantum non-signalling correlations, and formulated this problem as a semidefinite program depending only on the Kraus operator space of the channel. For the class of classical-quantum channels, they showed that the asymptotic zero-error classical capacity assisted by quantum non-signalling corre...

Let f be a Bianchi modular form, that is, an automorphic form for GL2 over an imaginary quadratic field F . In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each prime p of F above p we prove the existence of an L-invariant Lp, depending only on p and f , such that when the p-adic L-function of f has an exceptional zero at...

Let R be a commutative ring with nonzero identity, Z(R) be its set of zero-divisors, Nil(R) be its ideal of nilpotent elements, and U(R) be its group of units. We define a nonempty proper subset H of R to be a multiplicative-prime subset of R if the following two conditions hold: (i) ab ∈ H for every a ∈ H and b ∈ R; (ii) if ab ∈ H for a, b ∈ R, then either a ∈ H or b ∈ H . For example, H is mu...