On rings whose number of centralizers of ideals is finite

LetR be a ring. For the setF of all nonzero ideals ofR, we introduce an equivalence relation inF as follows: For idealsI andJ, I～J if and only ifV _R (I)=V _R(J), whereV _R() is the centralizer inR. LetI _R=F/～. Then we can see thatn(I _R), the cardinality ofI _R, is 1 if and only ifR is either a prime ring or a commutative ring (Theorem 1.1). An idealI ofR is said to be a commutator ideal ifI is generated by{st?ts; s∈S, t∈T} for subsetS andT ofR, andR is said to be a ring with (N) if any commutator ideal contains no nonzero nilpotent ideals. Then we have the following main theorem: LetR be a ring with (N). Thenn(I _R) is finite if and only ifR is isomorphic to an irredundant subdirect sum ofS⊕Z whereS is a finite direct sum of non commutative prime rings andZ is a commutative ring (Theorem 2.1). Finally, we show that the existence of a ringR such thatn(I _R)=m for any given natural numberm.     

A sublattice of the Leech lattice

In this paper a rank 12 even lattice C is constructed which is type 3 (if v ϵ C then (v, v) ⩾ 6), and which has the following maximality property: C ⊕ C + 3 · Λ₂₄ can be embedded in the Leech lattice Λ₂₄, and is a maximal type 3 sublattice of Λ₂₄. The construction uses properties of the binary and ternary Golay codes.     

Lie algebras whose lattice of ideals is closely related with a subspace lattice

Relationships between the structure of a Lie algebra and that of its lattice of ideals is studied for those Lie algebras whose ideal lattice is very close to that of an almost-abelian Lie algebra. It is shown here that if the base field is algebraically closed, finite or the real one, for any n ≥3 the only solvable Lie algebra whose lattice of ideals is isomorphic to that of the (n+l)-dimensional almost-abelian Lie algebra is itself.     

The Frattini sublattice of a finite distributive lattice

Frattini sublattices of finite distributive lattices are characterized and several applications are given thereof.Presented by J. Berman.The support of Consejo Nacional de Investigaciones Cientificas y Técnicas de la República Argentina is gratefully acknowledged.     

Epigroups whose subepigroup lattice is lower semimodular

We characterize epigroups mentioned in the title.     

The lattice of closure operators on a subgroup lattice

We say a lattice L is a subgroup lattice if there exists a group G such that Sub(G)?L, where Sub(G) is the lattice of subgroups of G, ordered by inclusion. We prove that the lattice of closure operators which act on the subgroup lattice of a finite group G is itself a subgroup lattice if and only if G is cyclic of prime power order.     

Groups in which every non-abelian subgroup is permutable

A subgroupH of a groupG is said to bepermutable ifHX=XH for every subgroupX ofG. In this paper the structure of groups in which every subgroup either is abelian or permutable is investigated. This work was done while the last author was visiting the University of Napoli Federico II. He thanks the “Dipartimento di Matematica e Applicazioni” for its financial support.     

Groups with elementary Abelian centralizers of involutions

An involution i of a group G is said to be almost perfect in G if any two involutions of i^G the order of a product of which is infinite are conjugated via a suitable involution in i^G. We generalize a known result by Brauer, Suzuki, and Wall concerning the structure of finite groups with elementary Abelian centralizers of involutions to groups with almost perfect involutions. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 75–82, January–February, 2007.     

On a sublattice of the lattice of congruences on a θ-regular semigroup

Let S be a regular semigroup. The lattice of all idempotent-separating congruences on S and the lattice of all group congruences on S are both modular sublattices of the full lattice of congruences on S. It is evident that the set theoretical union of these two sublattices, (S), is also a sublattice of the full lattice of congruences on S. It is natural to ask: Under what conditions is the sublattice (S) modular? In this paper we obtain a necessary and sufficient condition for the sublattice (S) to be modular when S is what we call a θ-regular semigroup. Bisimple ω-semigroups and simple regular ω-semigroups are θ-regular semigroups and so this paper extends the work of Munn [5] and Baird [1].