A Note on Jordan Triple Higher *-Derivations on Semiprime Rings

We introduce the following notion. Let ℕ0 be the set of all nonnegative integers and let D=(di)i∈ℕ0 be a family of additive mappings of a *-ring R such that d0=idR; D is called a Jordan higher *-derivation (resp., a Jordan higher *-derivation) of R if dn(x2)=∑i+j=n‍di(x)dj(x*i) (resp., dn(xyx)=∑i+j+k=n‍di(x)dj(y*i)dk(x*i+j)) for all x,y∈R and each n∈ℕ0. It is shown that the notions of Jordan higher *-derivations and Jordan triple higher *-derivations on a 6-torsion free semiprime *-ring are coincident.

When Is the Complement of the Comaximal Graph of a Commutative Ring Planar?

Let R be a commutative ring with identity. In this paper we classify rings R such that the complement of comaximal graph of R is planar. We also consider the subgraph of the complement of comaximal graph of R induced on the set S of all nonunits of R with the property that each element of S is not in the Jacobson radical of R and classify rings R such that this subgraph is planar.

On 10-Centralizer Groups of Odd Order

Let G be a group, and let Cent(G) denote the number of distinct centralizers of its elements. A group G is called n-centralizer if Cent(G)=n. In this paper, we investigate the structure of finite groups of odd order with Cent(G)=10 and prove that there is no finite nonabelian group of odd order with Cent(G)=10.

Introduction to gb-Triple Systems

This paper introduces the category of gb-triple systems and studies some of their algebraic properties. Also provided is a functor from this category to the category of Leibniz algebras.

An Investigation on Algebraic Structure of Soft Sets and Soft Filters over Residuated Lattices

We introduce the notion of soft filters in residuated lattices and investigate their basic properties. We investigate relations between soft residuated lattices and soft filter residuated lattices. The restricted and extended intersection (union), ∨ and ∧-intersection, cartesian product, and restricted and extended difference of the family of soft filters residuated lattices are established. Also, we consider the set of all soft sets over a universe set U and the set of parameters P with respect to U, SoftP(U), and we study its structure.

Divisibility Properties of the Fibonacci, Lucas, and Related Sequences

We use matrix techniques to give simple proofs of known divisibility properties of the Fibonacci, Lucas, generalized Lucas, and Gaussian Fibonacci numbers. Our derivations use the fact that products of diagonal matrices are diagonal together with Bezout’s identity.

On Generalized Jordan Triple (σ,τ)-Higher Derivations in Prime Rings

Let R be a ring and let U be a Lie ideal of R. Suppose that σ,τ are endomorphisms of R, and ℕ is the set of all nonnegative integers. A family F={fn}n∈ℕ of mappings fn:R→R is said to be a generalized (σ,τ)-higher derivation (resp., generalized Jordan triple (σ,τ)-higher derivation) of R if there exists a (σ,τ)-higher derivation D={dn}n∈ℕ of R such that f0=IR, the identity map on R, fn(a+b)=fn(a)+fn(b), and fn(ab)=∑i+j=nfi(σn-i(a))dj(τn-j(b)) (resp., fn(aba)=∑i+j+k=nfi(σn-i(a))dj(σkτi(b))dk(τn-k(a))) hold for all a,b∈R and for every n∈ℕ. If the above conditions hold for all a,b∈U, then F is said to be a generalized (σ,τ)-higher derivation (resp., generalized Jordan triple (σ,τ)-higher derivation) of U into R. In the present paper it is shown that if U is a noncentral square closed Lie ideal of a prime ring R of characteristic different from two, then every generalized Jordan triple (σ,τ)-higher derivation of U into R is a generalized (σ,τ)-higher derivation of U into R.

Generalized ⊕-Radical Supplemented Modules

Çalışıcı and Türkmen called a module M generalized ⊕-supplemented if every submodule has a generalized supplement that is a direct summand of M. Motivated by this, it is natural to introduce another notion that we called generalized ⊕-radical supplemented modules as a proper generalization of generalized ⊕-supplemented modules. In this paper, we obtain various properties of generalized ⊕-radical supplemented modules. We show that the class of generalized ⊕-radical supplemented modules is closed under finite direct sums. We attain that over a Dedekind domain a module M is generalized ⊕-radical supplemented if and only if M/P(M) is generalized ⊕-radical supplemented. We completely determine the structure of these modules over left V-rings. Moreover, we characterize semiperfect rings via generalized ⊕-radical supplemented modules.

Ioana's Superrigidity Theorem and Orbit Equivalence Relations

We give a survey of Adrian Ioana's cocycle superrigidity theorem for profinite actions of Property (T) groups and its applications to ergodic theory and set theory in this expository paper. In addition to a statement and proof of Ioana's theorem, this paper features the following: (i) an introduction to rigidity, including a crash course in Borel cocycles and a summary of some of the best-known superrigidity theorems; (ii) some easy applications of superrigidity, both to ergodic theory (orbit equivalence) and set theory (Borel reducibility); and (iii) a streamlined proof of Simon Thomas's theorem that the classification of torsion-free abelian groups of finite rank is intractable.

The EA-Dimension of a Commutative Ring

An elementary annihilator of a ring A is an annihilator that has the form (0:a)A; a∈R∖(0). We define the elementary annihilator dimension of the ring A, denoted by EAdim(A), to be the upper bound of the set of all integers n such that there is a chain (0:a0)⊂⋯⊂(0:an) of annihilators of A. We use this dimension to characterize some zero-divisors graphs.