Redefined generalized fuzzy ideals of near-rings

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.

On Commutativity of Semiprime Right Goldie C<i>_k</i>-Rings

This short exposition is about some commutativity conditions on a semiprime right Goldie Ck-ring. In particular, it is observed here that a semiprime right Goldie Ck-ring with symmetric quotient is commutative. The statement holds if the symmetric ring is replaced by reduced, 2-primal, left duo, right duo, abelian, NI, NCI, IFP, or Armendariz ring.

A Parallel Probabilistic Approach to Factorize a Semiprime

In accordance with the distributive traits of semiprimes’ divisors, the article proposes an approach that can find out the small divisor of a semiprime by parallel computing. The approach incorporates a deterministic search with a probabilistic search, requires less memory and can be implemented on ordinary multicore computers. Experiments show that certain semiprimes of 27 to 46 decimal-bits can be validly factorized with the approach on personal computer in expected time.

On Finite Rank Operators on Centrally Closed Semiprime Rings

We prove that the multiplication ring of a centrally closed semiprime ring R has a finite rank operator over the extended centroid C iff R contains an idempotent q such that qRq is finitely generated over C and, for each , there exist and e an idempotent of C such that xz=eq.

Characterizations of Hemirings by the Properties of Their <i>k</i>-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.

f-Algebra in Quotient Space

Based on the knowledge off-algebra, the concept of quotient f-algebra is mainly discussed. The sufficient condition when the quotient space is quotient f-algebra is given, and the quotient f-algebra that makes the kernel of a Riesz homomorphism as the equivalence class is obtained. Moreover, some important algebraic properties, such as commutative, semiprime and unital are studied, and the relationship between an f-algebra and its quotient space with respect to these properties are discussed in details.

(Right)Partial Generalized Automorphisms of Semiprime Rings

In this paper, we define the concept of (right) partial generalized automorphisms and discuss the extension problem, and also give a characterization of (right) partial generalized automorphisms of semiprime rings. Finally, we study the centralizing problem of right partial generalized automorphisms.

Nilpotent Elements and Nil-Reflexive Property of Generalized Power Series Rings

Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized power series reflexive and nil generalized power series reflexive, respectively. We obtain various necessary or sufficient conditions for a ring to be generalized power series reflexive and nil generalized power series reflexive. Examples are given to show that, nil generalized power series reflexive need not be generalized power series reflexive and vice versa, and nil generalized power series reflexive but not semicommutative are presented. We proved that, if R is a left APP-ring, then R is generalized power series reflexive, and R is nil generalized power series reflexive if and only if R/I is nil generalized power series reflexive. Moreover, we investigate ring extensions which have roles in ring theory.

Generalized Derivations with Nilpotent Values on Semiprime Rings

Let R be a semiprime ring, R F be its left Martindale quotient ring and I be an essential ideal of R. Then every generalized derivation μ defined on I can be uniquely extended to a generalized derivation of R F . Furthermore, if there exists a fixed positive integer n such that μ(x) n = 0 for all x ∈I, then μ = 0.

A Note on Commutators with Power Central Values on Lie Ideals

This note proves that, if R is a prime ring of characteristic 2 with d a derivation of R and L a noncentral Lie ideal of R such that [d（u）,u]^n is central, for all u ∈ L, then R must satisfy s4, the standard identity in 4 variables. The case where R is a semiprime ring is also examined by the authors. The results of the note improve Carini and Filippis＇s results.

On Generalized （m, n）-Derivations and Generalized （m, n）-Jordan Derivations in Rings

The aim of this paper is to define the notions of generalized （m, n）-derivations and generalized （m, n）：Jordan derivations and to prove two theorems involving these map- pings.

On Certain Functional Equation in Semiprime Rings

Let n be a fixed integer, let R be an （n ＋ 1）!-torsion free semiprime ring with the identity element and let F ： R → R be an additive mapping satisfying the relation F（xn＋2） ∑＋n＋i=1xi-1F（x）2xn＋1-i -∑xnF（x）xn＋1-i for all x 6 R. In this case, we prove that F is of the form 2F（x） = D（x） ＋ ax ＋ xa for all x ∈ R, where D ： R → R is a derivation and a 6 R is some fixed element.

On ε-Derivations and Local ε-Derivations

In this paper, we describe ∈-derivations in certain graded algebras by their actions on elements satisfying some special conditions. One of the main results is applied to local ∈-derivations on some certain graded algebras.

Valued Gabriel quiver of a wedge product and semiprime coalgebras

We describe the valued Gabriel quiver of a wedge product of coalgebras and study the category of comodules of a semiprime coalgebra. In particular, we prove that any monomial semiprime k-tame fc-tame coalgebra is string. We also prove a version of Eisenbud-Griffith theorem for coalgebras, namely, any hereditary semiprime strictly quasi-finite coalgebra is serial.

Orthogonality of Generalized (θ,φ)-Derivations on Ideals

In this paper,necessary and sufficient conditions concerning the orthogonality and the composition of a couple of generalized （θ,φ）-derivations on a nonzero ideal of a semiprime ring are presented.These results are generalizations of several results of Breˇsar and Vukman,which are related to a theorem of Posner on the product of two derivations on a prime ring.