This paper proposes the novel algebraic structure of a linear ring space. A linear ring space is an order triad consisting of two rings, and a linear map between the two rings. The definition of quasi-linearity is dis...This paper proposes the novel algebraic structure of a linear ring space. A linear ring space is an order triad consisting of two rings, and a linear map between the two rings. The definition of quasi-linearity is discussed, in addition to the examination of properties and classifications of linear ring spaces. Particularly, the ring of holomorphic functions on a region of the complex plane is examined, and the manner in which it generates an iterated linear ring space under the complex derivative operator. This notion is then generalized to all rings with nth order linear and surjective operators. Basic operator theory regarding the classifications of linear ring maps is also covered.展开更多
Let R be a commutative ring with identity and M an R-module. In this paper, we relate a graph to M, say Γ(M), provided tsshat when M=R, Γ(M)is exactly the classic zero-divisor graph.
We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R...We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R)[X]=W(R)[X]■Z(R[X]),where J(A),N^(*)(A),W(A),Z(A)are the Jacobson radical,upper nilradical,Wedderburn radical,and center of a given ring A,respectively,and A[X]denotes the polynomial ring with a set X of commuting indeterminates over A;we also prove that R is semiprime if and only if the right(left)singular ideal of R is zero.We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular,from any given ring.Moreover,for a C-regular ring R,the following are proved to be equivalent:(i)R is Abelian;(ii)every prime factor ring of R is a duo domain;(ii)R is quasi-duo;and(iv)R/W(R)is reduced.展开更多
In this paper we show that if R is a discrete valuation ring, then R is a filtered ring. We prove some properties and relation when R is a discrete valuation ring.
Fog computing is a rapidly growing technology that aids in pipelining the possibility of mitigating breaches between the cloud and edge servers.It facil-itates the benefits of the network edge with the maximized probab...Fog computing is a rapidly growing technology that aids in pipelining the possibility of mitigating breaches between the cloud and edge servers.It facil-itates the benefits of the network edge with the maximized probability of offering interaction with the cloud.However,the fog computing characteristics are suscep-tible to counteract the challenges of security.The issues present with the Physical Layer Security(PLS)aspect in fog computing which included authentication,integrity,and confidentiality has been considered as a reason for the potential issues leading to the security breaches.In this work,the Octonion Algebra-inspired Non-Commutative Ring-based Fully Homomorphic Encryption Scheme(NCR-FHE)was proposed as a secrecy improvement technique to overcome the impersonation attack in cloud computing.The proposed approach was derived through the benefits of Octonion algebra to facilitate the maximum security for big data-based applications.The major issues in the physical layer security which may potentially lead to the possible security issues were identified.The potential issues causing the impersonation attack in the Fog computing environment were identified.The proposed approach was compared with the existing encryption approaches and claimed as a robust approach to identify the impersonation attack for the fog and edge network.The computation cost of the proposed NCR-FHE is identified to be significantly reduced by 7.18%,8.64%,9.42%,and 10.36%in terms of communication overhead for varying packet sizes,when compared to the benchmarked ECDH-DH,LHPPS,BF-PHE and SHE-PABF schemes.展开更多
In this paper, we obtain a new kind of complete Lie algebra over a commutative ring, which is the Lie algebra consisting of all n × n anti-symmetric matrices over a 2-torsionfree commutative ring with identity.
This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated...This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R)is a commutative regular ring with J(R)nil,where J(R)is the Jacobson radical of R.展开更多
The aim of this paper is to investigate higher level orderings on modulesover commutative rings. On the basis of the theory of higher level orderings on fields andcommutative rings, some results involving existence of...The aim of this paper is to investigate higher level orderings on modulesover commutative rings. On the basis of the theory of higher level orderings on fields andcommutative rings, some results involving existence of higher level orderings are generalized to thecategory of modules over commutative rings. Moreover, a strict intersection theorem for higherlevel orderings on modules is established.展开更多
For a commutative ring with identity, we obtain a complete description of all overgroups of unitary groups U2nR (n ≥ 5), which include symplectic, ordinary orthogonal and standard unitary groups, in linear group GL2nR.
Let L be the symplectic algebra or the orthogonal algebra over a commutative ring R, h the maximal torus of L consisting of all diagonal matrices in L, and b the standard Borel subalgebra of L containing h. In this pa...Let L be the symplectic algebra or the orthogonal algebra over a commutative ring R, h the maximal torus of L consisting of all diagonal matrices in L, and b the standard Borel subalgebra of L containing h. In this paper, we first determine the intermediate algebras between h and b, then for such an intermediate algebra, we give an explicit description on its derivations, provided that R is a commutative ring with identity and 2 is invertible in R.展开更多
文摘This paper proposes the novel algebraic structure of a linear ring space. A linear ring space is an order triad consisting of two rings, and a linear map between the two rings. The definition of quasi-linearity is discussed, in addition to the examination of properties and classifications of linear ring spaces. Particularly, the ring of holomorphic functions on a region of the complex plane is examined, and the manner in which it generates an iterated linear ring space under the complex derivative operator. This notion is then generalized to all rings with nth order linear and surjective operators. Basic operator theory regarding the classifications of linear ring maps is also covered.
文摘Let R be a commutative ring with identity and M an R-module. In this paper, we relate a graph to M, say Γ(M), provided tsshat when M=R, Γ(M)is exactly the classic zero-divisor graph.
文摘We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R)[X]=W(R)[X]■Z(R[X]),where J(A),N^(*)(A),W(A),Z(A)are the Jacobson radical,upper nilradical,Wedderburn radical,and center of a given ring A,respectively,and A[X]denotes the polynomial ring with a set X of commuting indeterminates over A;we also prove that R is semiprime if and only if the right(left)singular ideal of R is zero.We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular,from any given ring.Moreover,for a C-regular ring R,the following are proved to be equivalent:(i)R is Abelian;(ii)every prime factor ring of R is a duo domain;(ii)R is quasi-duo;and(iv)R/W(R)is reduced.
基金Supported by National Natural Science Foundation of China(1112612111426093)+3 种基金Doctor Foundation of Henan Polytechnic University(B2010-93)Natural Science Research Program of Science and Technology Department of Henan Province(112300410120)Natural Science Research Program of Education Department of Henan Province(2011B110016)Applied Mathematics Provincial-level Key Discipline of Henan Province
文摘In this paper we show that if R is a discrete valuation ring, then R is a filtered ring. We prove some properties and relation when R is a discrete valuation ring.
文摘Fog computing is a rapidly growing technology that aids in pipelining the possibility of mitigating breaches between the cloud and edge servers.It facil-itates the benefits of the network edge with the maximized probability of offering interaction with the cloud.However,the fog computing characteristics are suscep-tible to counteract the challenges of security.The issues present with the Physical Layer Security(PLS)aspect in fog computing which included authentication,integrity,and confidentiality has been considered as a reason for the potential issues leading to the security breaches.In this work,the Octonion Algebra-inspired Non-Commutative Ring-based Fully Homomorphic Encryption Scheme(NCR-FHE)was proposed as a secrecy improvement technique to overcome the impersonation attack in cloud computing.The proposed approach was derived through the benefits of Octonion algebra to facilitate the maximum security for big data-based applications.The major issues in the physical layer security which may potentially lead to the possible security issues were identified.The potential issues causing the impersonation attack in the Fog computing environment were identified.The proposed approach was compared with the existing encryption approaches and claimed as a robust approach to identify the impersonation attack for the fog and edge network.The computation cost of the proposed NCR-FHE is identified to be significantly reduced by 7.18%,8.64%,9.42%,and 10.36%in terms of communication overhead for varying packet sizes,when compared to the benchmarked ECDH-DH,LHPPS,BF-PHE and SHE-PABF schemes.
文摘In this paper, we obtain a new kind of complete Lie algebra over a commutative ring, which is the Lie algebra consisting of all n × n anti-symmetric matrices over a 2-torsionfree commutative ring with identity.
基金The second author was supported by the National Research Foundation of Korea(NRF)grant funded by the Korea government(MSIT)(No.2019R1F1A1040405).
文摘This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R)is a commutative regular ring with J(R)nil,where J(R)is the Jacobson radical of R.
基金1)Project supported hy the National Natural Science Foundation of China,Grant No,19661002
文摘The aim of this paper is to investigate higher level orderings on modulesover commutative rings. On the basis of the theory of higher level orderings on fields andcommutative rings, some results involving existence of higher level orderings are generalized to thecategory of modules over commutative rings. Moreover, a strict intersection theorem for higherlevel orderings on modules is established.
基金supported by the National Natural Science Foundation(Grant No.10571033)the Research Fund for the Doctoral of Higher Education of China(Grant No.20040213006)Cultivation Fund of the Key Scientific and Technical Innovation Project Ministry of Education of China(Grant No.704004).
文摘For a commutative ring with identity, we obtain a complete description of all overgroups of unitary groups U2nR (n ≥ 5), which include symplectic, ordinary orthogonal and standard unitary groups, in linear group GL2nR.
文摘Let L be the symplectic algebra or the orthogonal algebra over a commutative ring R, h the maximal torus of L consisting of all diagonal matrices in L, and b the standard Borel subalgebra of L containing h. In this paper, we first determine the intermediate algebras between h and b, then for such an intermediate algebra, we give an explicit description on its derivations, provided that R is a commutative ring with identity and 2 is invertible in R.
