Commutative rings whose ideal lattices are complemented

It is well known that the set of all ideals(2) of a ring forms a complete modular lattice with respect to set inclusion. The same is true of the set of all right ideals. Our purpose in this paper is to consider the consequences of imposing certain additional restrictions on these ideal lattices. In particular, we discuss the case in which one or both of these lattices is complemented, and the case in which one or both is distributive. In §1 two strictly latticetheoretic results are noted for the sake of their application to the complemented case. In §2 rings which have a complemented ideal lattice are considered. Such rings are characterized as discrete direct sums of simple rings. The structure space of primitive ideals of such rings is also discussed. In §3 corresponding results are obtained for rings whose lattice of right ideals is complemented. In particular, it is shown that a ring has a complemented right ideal lattice if and only if it is isomorphic with a discrete direct sum of quasi-simple rings. The socle [7](3) and the maximal regular ideal [5] are discussed in connection with such rings. The effect of an identity element is considered in §4. In §5 rings with distributive ideal lattices are considered and still another variant of regularity [20] is introduced. It is shown that a semi-simple ring with a distributive right ideal lattice is isomorphic with a subdirect sum of division rings. In the concluding section a type of ideal, introduced by L. Fuchs [9] in connection with commutative rings with distributive ideal lattice, and which we call strongly irreducible, is considered. Some properties of these ideals, analogous to corresponding ones for prime ideals [19], are developed. Finally, it is observed that a topology may be introduced in the set of all proper strongly irreducible ideals in such a way that the resulting space contains the spaces of prime [19] and primitive [13] ideals as subspaces. I wish to take this opportunity to thank Professor M. F. Smiley for the encouragement and many helpful suggestions he has given me throughout the preparation of this paper. 1. Some lattice-theoretic preliminaries. In this section we state two results of a strictly lattice-theoretic nature with a view toward subsequent applications to rings. Our notation and terminology is that of Birkhoff [3]. In

Well-rounded twists of ideal lattices from real quadratic fields

The proliferation of cloud computing has rendered secure data sharing a paramount challenge, particularly in the post-quantum era where quantum computers threaten traditional public-key cryptosystems. While many schemes exist, achieving fine-grained, efficient, and quantum-resistant access control remains a key research area. To this end, this paper proposes a post-quantum secure data sharing scheme based on ideal lattices. We construct a new Ciphertext Policy Attribute-Based Encryption (CP-ABE) scheme for key distribution over ideal lattices, integrated into a hybrid encryption mechanism to protect data privacy efficiently. This approach allows data owners to enforce complex access policies on their encrypted data, ensuring that only authorized users can access it. Rigorous security proofs demonstrate that the proposed scheme is IND-CPA-secure under the Ring-LWE assumption. The efficiency of the scheme is due to its effective utilization of the algebraic structure of the ideal lattice, which accelerates polynomial operations through Number Theoretic Transformations (NTT). Comparative analysis and experimental results show that the scheme has significant advantages in computing efficiency and storage cost, thereby underscoring its practical feasibility for high-throughput cloud environments.

By electric transport measurements on high-purity bulk molybdenum single crystals at helium-temperature, it is shown that when approaching the ideal metallic crystal structure, there are deviations from Ohm’s law that can exceed 30% even at low current densities ≅3A/mm2. This is due to the intrinsic circular magnetic field of the measuring current in the range of B≅1mT, which generates a magnetoresistance according to the two-band conduction model. With increasing lattice vibrations and/or crystal defects, this inherent uncertainty finally reduces to immeasurably small values. Finally, Ohm’s law for a nearly ideal lattice structure can be strictly exact only in the theoretical limiting case of an infinite lattice plane, because then the magnetic fields of all electrons of the current compensate to zero.

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G T=<1>,where G T is the inertial group of R.

The fundamental relation on a fuzzy hyperring is de-fined as the smallest equivalence relation, such that the quotient would be the ring, that is not commutative necessarily. In this pa-per, we introduce a new fuzzy strongly regular equivalence on fuzzy hyperrings, where the ring is commutative with respect to both sum and product. With considering this relation on fuzzy hyperring, the set of the quotient is a commutative ring. Also, we introduce fundamental functor between the category of fuzzy hyperrings and category of commutative rings and some related properties. Even-tually, we introduce α-part in fuzzy hyperring and determine some necessary and sufficient conditions so that the relation α is transi-tive.

A commutative ring R is said to be strongly Hopfian if the chain of annihilators ann(a) ⊆ ann(a 2) ⊆ … stabilizes for each a ∈ R. In this article, we are interested in the class of strongly Hopfian rings and the transfer of this property from a commutative ring R to the ring of the power series R[[X]]. We provide an example of a strongly Hopfian ring R such that R[[X]] is not strongly Hopfian. We give some necessary and sufficient conditions for R[[X]] to be strongly Hopfian.

We give a constructive sufficient condition for a matrix over a commutative ring to be von Neumann regular, and we show that it is also necessary over local rings. Specifically, we prove that a matrix A over a local commutative ring is von Neumann regular if and only if A has an invertible -submatrix if and only if the determinantal rank and the McCoy rank of A coincide. We deduce consequences to (products of local) commutative rings, and we determine the number of von Neumann regular matrices over some finite rings of residue classes and group algebras.

Preserving pole assignable systems and reachable systems in commutative rings

Sufficient conditions for a quantum dynamical semigroup (QDS) to be unital are proved for a class of problems in quantum optics with Hamiltonians which are self-adjoint polynomials of any finite order in creation and annihilation operators. The order of the Hamiltonian may be higher than the order of completely positive part of the formal generator of a QDS.

A class of von Neumann algebras associated with the normal representation (i.e. the Fock representation) of the canonical commutation relations has been studied in an earlier paper. > The von Neumann algebra in question is always the tensor product of an abelian algebra and a factor of the infinite type. A necessary and sufficient condition for the factor to be of type Icc has been obtained in reference 1). In the present paper it will be shown that the factor in question is of type IIIcc unless it is of type t'. Recently there have been some interests in the type of von Neumann algebras of local observables in quantum field theory. J,SJ,), ) The von Neumann algebras of 'local observables for a free scalar field) as well as those for a generalized free field belong to the class of von Neumann algebras considered in reference 1) and our result implies that they are either type III' or t:o. In particular, for those cases where type Icc have been excluded,J,sl, l the algebra must be of type IIICXJ . Our result also implies that the algebra of all creation and annihilation operators in quantum theory of an infinite free Bose gasl is a factor of type IIIcc when no condensation is present.

Let R be a commutative ring with multiplicative identity and P is a finitely generated projective R-module. If P∗ is the set of R-module homomorphism from P to R, then the tensor product P∗⊗RP can be considered as an R-coalgebra. Furthermore, P and P∗ is a comodule over coalgebra P∗⊗RP. Using the Morita context, this paper give sufficient conditions of clean coalgebra P∗⊗RP and clean P∗⊗RP-comodule P and P∗. These sufficient conditions are determined by the conditions of module P and ring R.

The invertibility of linear systems with incommensurable delays which are looked upon as systems over commutative rings is considered in this paper. A new notion of k -integral and \tau -delay inverse for these systems is introduced. In respect of this key notion, necessary and sufficient conditions are established to ensure the existence of an inverse for the above systems.

The condition that certain left ideals in a finite monoid generate projective left ideals in the semigroup algebra imposes a strong restriction on the intersection of principal left ideals in the semigroup. Let S be a finite monoid, k be a commutative ring with identity, and let Ic S be a left ideal in S. We demand that kI be projective as a left kSmodule and investigate the resulting restrictions on the structure of I. In particular we can look for necessary conditions on S for kS to be left hereditary. (A sufficient condition is obtained in [4] and [5].) Semigroup terminology below follows [1] and [2]. We first need the following facts, which are valid in any ring with identity. LEMMA 1. Let R be a ring with identity. Let Ic R be a left ideal which is projective as a left R-module. Let e E R be any idempotent. Then the left ideal I+ Re is projective if and only if JrflRe is a direct summand of L PROOF. We observe that we have the following two short exact sequences: 0 I(1 e) -* I + Re Re > 0, 0OI n Re-I--I I(1 -e)->*O, where the map on the right end of the first sequence is xH-*xe, which has kernel (I+Re) r)R(1 -e)=I(1 -e), and the map on the right end of the second sequence is xF-*x(1 -e). Since Re is projective, the first sequence always splits, so that I+Re is projective if and only if I(1 -e) is. On the other hand, since I is projective, 1(1 -e) is projective if and only if the second sequence splits. Received by the editors June 5, 1972. AMS (MOS) subject classifications (1970). Primary 20M10, 20M25; Secondary 16A32, 16A50, 16A60.

Given a commutative ring R R , we investigate the structure of the set of Artinian subrings of R R . We also consider the family of zero-dimensional subrings of R R . Necessary and sufficient conditions are given in order that every zero-dimensional subring of a ring be Artinian. We also consider closure properties of the set of Artinian subrings of a ring with respect to intersection or finite intersection, and the condition that the set of Artinian subrings of a ring forms a directed family.