Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp. R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions of local cohomology modules, and show that if the Lynch's conjecture, i...

We show a novel lattice-based scheme (PairTRU) which is a non-commutative variant of the NTRU. The original NTRU is defined via the ring of quotient with variable in integers and this system works in the ring R = Z[x] . We extend this system over Z× Z and it performs all of operations in the non-commutative ring M = M(k,Z×Z)[x] <(Ik×k,Ik×k)x−(Ik×k,Ik×k)> , where M is a matrix ring of k ×...

A commutative local Cohen-Macaulay ring R of finite Cohen-Macaulay type is known to be an isolated singularity; that is, Spec(R) \ {m} is smooth. This paper proves a non-commutative analogue. Namely, if A is a (non-commutative) graded Artin-Schelter Cohen-Macaulay algebra which is FBN and has finite Cohen-Macaulay type, then the non-commutative projective scheme determined by A is smooth.

Abstract The theory of R. Crowell on derived modules is approached within the non-commutative differential modules. We also seek analogies to cotangent complex from differentials in commutative ring setting. Finally, we give examples motivated Galois coverings curves.

In this paper, we propose a new method for designing public key cryptosystems based on general non-commutative rings. The key idea of our proposal is that for a given non-commutative ring, we can define polynomials and take them as the underlying work structure. By doing so, it is easy to implement Diffie-Helman-like key exchange protocol. And consequently, ElGamal-like cryptosystems can be der...

Authentication is a process by which an entity, which could be a person or intended computer, establishes its identity to another entity. In private and public computer networks including the Internet, authentication is commonly done through the use of logon passwords. Knowledge of the password is assumed to guarantee that the user is authentic. Internet business and many other transactions req...

The rings considered in this article are commutative with identity which are not fields. Let R be a ring. A. Alilou, J. Amjadi and Sheikholeslami introduced and investigated a graph whose vertex set is the set of all nontrivial ideals of R and distinct vertices I, J are joined by an edge in this graph if and only if either ann(I)J = (0) or ann(J)I = (0). They called this graph as a new graph as...

Abstract The Learning with Errors (LWE) problem is the fundamental backbone of modern lattice-based cryptography, allowing one to establish cryptography on hardness well-studied computational problems. However, schemes based LWE are often impractical, so Ring was introduced as a form ‘structured’ LWE, trading off hard quantify loss security for an increase in efficiency by working over well-cho...