A topological space denoted by Fspec(D),called L-fuzzy prime spectrum of a bounded distributive lattice D is introduced.This space Fspec(D) is compact and it contains a subspace homeomorphic with the prime spectrum of...A topological space denoted by Fspec(D),called L-fuzzy prime spectrum of a bounded distributive lattice D is introduced.This space Fspec(D) is compact and it contains a subspace homeomorphic with the prime spectrum of D which is dense in it.The correspondence associating D to the topological space Fspec(D) is shown to define a contravarient functor from the category of bounded distributive lattices into the category of compact topological spaces.展开更多
A Noetherian(Artinian)Lie algebra satisfies the maximal(minimal)condition for ideals.Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras.We study conditions on prime ideals relating these properti...A Noetherian(Artinian)Lie algebra satisfies the maximal(minimal)condition for ideals.Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras.We study conditions on prime ideals relating these properties.We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals,and an ideally finite Lie algebra is quasi-Noetherian if and only if it is quasi-Artinian.Both properties are equivalent to soluble-by-finite.We also prove a structure theorem for serially finite Artinian Lie algebras.展开更多
Let R be a prime ring with an automorphism σ≠1, an identity map. Let L be a noncentral Lie ideal of R such that \xσ, x] ∈Z for all x ∈ L, where Z is the center of R. Then L is contained in the center of R, unless...Let R be a prime ring with an automorphism σ≠1, an identity map. Let L be a noncentral Lie ideal of R such that \xσ, x] ∈Z for all x ∈ L, where Z is the center of R. Then L is contained in the center of R, unless char(R) = 2 and dimcRC = 4.展开更多
With a new idea, we redefine generalized fuzzy subnear-rings （ideals） of a near- ring and investigate some of its related properties. Some new characterizations are given. In particular, we introduce the concepts of...With a new idea, we redefine generalized fuzzy subnear-rings （ideals） of a near- ring and investigate some of its related properties. Some new characterizations are given. In particular, we introduce the concepts of strong prime （or semiprime） （∈, ∈∨ q）-fuzzy ideals of near-rings, and discuss the relationship between strong prime （or semiprime） （∈, ∈∨ q）-fuzzy ideals and prime （or semiprime） （∈, ∈∨ q）-fuzzy ideals of near-rings.展开更多
In this paper we characterize those hemirings for which each k-ideal is idempotent. We also characterize those hemirings for which each fuzzy k-ideal is idempotent. The space of prime k-ideals (fuzzy k-prime k-ideals)...In this paper we characterize those hemirings for which each k-ideal is idempotent. We also characterize those hemirings for which each fuzzy k-ideal is idempotent. The space of prime k-ideals (fuzzy k-prime k-ideals) is topologized.展开更多
Let d_i(1≤i≤n),δ_1,δ_2,δ_3 be nonzero derivations of a prime ring R with charR≠2.Suppose that U is a Lie ideal such that u^2∈U for all u∈U.In this paper,we prove that U(?) Z(R) when one of the following holds:...Let d_i(1≤i≤n),δ_1,δ_2,δ_3 be nonzero derivations of a prime ring R with charR≠2.Suppose that U is a Lie ideal such that u^2∈U for all u∈U.In this paper,we prove that U(?) Z(R) when one of the following holds:(1) d_1(x_1)d_2(x_2),...,d_n(x_n)∈Z(R)(2)δ_3(y)δ_1(x)=δ_2(x)δ_3(y).Further,if U is a Lie ideal and a subring then(3)δ_1(x)δ_2(y)+δ_2(x)δ_1(y)∈Z(R) for all x_i,x,y∈U.展开更多
Let R be a prime ring, L a noncentral Lie ideal and a nontrivialautomorphism of R such that us（u）ut = 0 for all u 2 L, where s; t are fixednon-negative integers. If either charR 〉 s ＋ t or charR = 0, then R satis...Let R be a prime ring, L a noncentral Lie ideal and a nontrivialautomorphism of R such that us（u）ut = 0 for all u 2 L, where s; t are fixednon-negative integers. If either charR 〉 s ＋ t or charR = 0, then R satisfies s4, thestandard identity in four variables. We also examine the identity （σ（[x; y]）-[x; y]）n =0 for all x; y ∈ I, where I is a nonzero ideal of R and n is a fixed positive integer. Ifeither charR 〉 n or charR = 0, then R is commutative.展开更多
Let R be a 2-torsion free prime ring and L a noncommutative Lie ideal of R. Suppose that （d,σ） is a skew derivation of R such that xsd（x）xt = 0 for all x ∈ L, where s, t are fixed non-negative integers. Then d = 0.
A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobson＇s famous result, several tech- niques are developed to achieve this goal. In the ...A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobson＇s famous result, several tech- niques are developed to achieve this goal. In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized （α β）-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.展开更多
基金UGC,New Delhi for financial support through scheme F.No33-109/2007(SR)
文摘A topological space denoted by Fspec(D),called L-fuzzy prime spectrum of a bounded distributive lattice D is introduced.This space Fspec(D) is compact and it contains a subspace homeomorphic with the prime spectrum of D which is dense in it.The correspondence associating D to the topological space Fspec(D) is shown to define a contravarient functor from the category of bounded distributive lattices into the category of compact topological spaces.
文摘A Noetherian(Artinian)Lie algebra satisfies the maximal(minimal)condition for ideals.Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras.We study conditions on prime ideals relating these properties.We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals,and an ideally finite Lie algebra is quasi-Noetherian if and only if it is quasi-Artinian.Both properties are equivalent to soluble-by-finite.We also prove a structure theorem for serially finite Artinian Lie algebras.
文摘Let R be a prime ring with an automorphism σ≠1, an identity map. Let L be a noncentral Lie ideal of R such that \xσ, x] ∈Z for all x ∈ L, where Z is the center of R. Then L is contained in the center of R, unless char(R) = 2 and dimcRC = 4.
基金Supported by the National Natural Science Foundation of China(60875034)the Natural Science Foundation of Education Committee of Hubei Province(D20092901),the Natural Science Foundation of Hubei Province(2009CDB340)
文摘With a new idea, we redefine generalized fuzzy subnear-rings （ideals） of a near- ring and investigate some of its related properties. Some new characterizations are given. In particular, we introduce the concepts of strong prime （or semiprime） （∈, ∈∨ q）-fuzzy ideals of near-rings, and discuss the relationship between strong prime （or semiprime） （∈, ∈∨ q）-fuzzy ideals and prime （or semiprime） （∈, ∈∨ q）-fuzzy ideals of near-rings.
文摘In this paper we characterize those hemirings for which each k-ideal is idempotent. We also characterize those hemirings for which each fuzzy k-ideal is idempotent. The space of prime k-ideals (fuzzy k-prime k-ideals) is topologized.
基金Supported by the Natural Science Research Item of Anhui Province College(KJ2008B013)
文摘Let d_i(1≤i≤n),δ_1,δ_2,δ_3 be nonzero derivations of a prime ring R with charR≠2.Suppose that U is a Lie ideal such that u^2∈U for all u∈U.In this paper,we prove that U(?) Z(R) when one of the following holds:(1) d_1(x_1)d_2(x_2),...,d_n(x_n)∈Z(R)(2)δ_3(y)δ_1(x)=δ_2(x)δ_3(y).Further,if U is a Lie ideal and a subring then(3)δ_1(x)δ_2(y)+δ_2(x)δ_1(y)∈Z(R) for all x_i,x,y∈U.
基金The NSF(1408085QA08) of Anhui Provincethe Natural Science Research Foundation(KJ2014A183) of Anhui Provincial Education DepartmentAnhui Province College Excellent Young Talents Fund Project(2012SQRL155) of China
文摘Let R be a prime ring, L a noncentral Lie ideal and a nontrivialautomorphism of R such that us（u）ut = 0 for all u 2 L, where s; t are fixednon-negative integers. If either charR 〉 s ＋ t or charR = 0, then R satisfies s4, thestandard identity in four variables. We also examine the identity （σ（[x; y]）-[x; y]）n =0 for all x; y ∈ I, where I is a nonzero ideal of R and n is a fixed positive integer. Ifeither charR 〉 n or charR = 0, then R is commutative.
基金The NSF(1408085QA08)of Anhui Provincialthe Key University Science Research Project(KJ2014A183)of Anhui Province of Chinathe Training Program(2014PY06)of Chuzhou University of China
文摘Let R be a 2-torsion free prime ring and L a noncommutative Lie ideal of R. Suppose that （d,σ） is a skew derivation of R such that xsd（x）xt = 0 for all x ∈ L, where s, t are fixed non-negative integers. Then d = 0.
基金Supported by the National Natural Science Foundation of China （No.40701179） and the Fundamental Research Funds for the Central Universities （No. XDJK2009C 102）.
文摘A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobson＇s famous result, several tech- niques are developed to achieve this goal. In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized （α β）-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.
