Neşet Aydin

• Çanakkale Onsekiz Mart University

In this paper, we introduce the notion of the one-sided generalized (α, β)−reversederivation of a ring R. Let R be a semiprime ring, ϱ be a non-zero ideal of R, α bean epimorphism of ϱ, β be a homomorphism of ϱ (α be a homomorphism of ϱ, βbe an epimorphism of ϱ) and γ : ϱ → R be a non-zero (α, β)−reverse derivation.We show that there exists F : ϱ →...

The study consists of two parts. The first part shows that if h1(x)h2(y) = h3(x)h4(y), for all x, y ∈ R, then h1 = h3 and h2 = h4. Here, h1, h2, h3, and h4 are zero-power valued non-zero homoderivations of a prime ring R. Moreover, this study provide an explanation related to h1 and h2 satisfying the condition ah1 + h2b = 0. The second part shows t...

In this study, the relationship between (α,β)--reverse derivation and (β,α)--derivation on an ideal of semiprime ring is investigated, which is here α and β are homomorphisms of the ideal. Herstein shown that if R is a prime ring and d is a nonzero reverse derivation of R, then R is commutative integral domain and d is a derivation. We transform He...

The algebraic properties and identities of a semiprime ring are investigated with the help of the multiplicative (generalised)-(α, α)-reverse derivation on the non-empty ideal of the semiprime ring.

The present paper investigates some properties of generalized reverse derivations on prime and semiprime rings. Firstly, the commutativity of a prime ring R is examined under the following differential identities provided by a generalized reverse derivation F associated with a reverse derivation d of R on a one-sided ideal of R and a mapping G: (i...

Let $R$ be a prime ring with multiplicative (generalized)-derivations $(F,f)$ and $(G,g)$ on $R$. This paper gives a number of central valued algebraic identities involving $F$ and $G$ that are equivalent to the commutativity of $R$ under some suitable assumptions. Moreover, in order to optimize our results, we show that the assumptions taken canno...

Let $R$ be a semiprime ring and $\alpha$ be an automorphism of $R.$ A mapping $F:R\to R$ (not necessarily additive) is called multiplicative generalized $(\alpha,\alpha)-$derivation if there exists a unique $(\alpha,\alpha)-$derivation $d$ of $R$ such that $F(xy)=F(x)\alpha(y)+\alpha(x)d(y)$ for all $x,y\in R.$ In the present paper, we intend to st...

The present paper deals with the commutativity of an associative ring R and a unital Banach Algebra A via derivations. Precisely, the study of multiplicative (generalized)-derivations F and G of semiprime (prime) ring R satisfying the identitiesG(xy) ± [F(x), y]± [x, y] ∈ Z(R) and G(xy) ± [x, F(y)]± [x, y] ∈ Z(R) has been carried out. Moreover, we...

Let R be a prime ∗-ring where ∗ be an involution of R, α be an automorphism of R, T be a nonzero left α-∗-centralizer on R and d be a nonzero ∗-α-derivation on R. The aim of this paper is to prove the commutativity of a ∗-ring R with the followings conditions: i) if T is a homomorphism (or an antihomomorphism) on R,ii) if d([x, y]) = 0 for all x, y...

In this study, we investigate commutavity of prime ring R with generalized reverse derivations F and G. Also, we proved that if L is square closed Lieideal, then L is contained in center Z (R) under given conditions in theorems.

Let R be a ∗−prime ring with characteristic not 2, U a nonzero ∗− ( σ,τ )−Lie ideal of R , d a nonzero derivation of R . Suppose σ , τ be two automorphisms of R such that σd = dσ , τd = dτ and ∗ commutes with σ , τ , d . In the present paper it is shown that if d ( U ) ⊆ Z or d² ( U ) ⊆ Z , then U ⊆ Z .

Let R be an associative ring. We define a subset SR of R as SR = fa 2 R j aRa = (0)g and call it the source of semiprimeness of R. We first examine some basic properties of the subset SR in any ring R, and then define the notions such as R being a jSRj-reduced ring, a jSRj-domain and a jSRj-division ring which are slight generalizations of their cl...

Leonardo Fibonacci 13. yy yaşamış İtalyan bir matematikçidir. Fibonacci için “Matematiği Araplar’dan alıp, Avrupa’ya aktaran kişi” denilebilir. Fibonacci yazdığı Liber Abaci’ya adlı kitabında yer alan bir problemde ortaya çıkan sayı dizisi ile tanınır. Bu dizi aşağıdaki gibidir:1,1,2,3,5,8,13,21,34,55,89,…Bu diziye bakıldığında basit bir kuralla ol...

Let R be a ∗-prime ring with characteristic not 2, σ,τ:R→R be two automorphisms, U be a nonzero ∗-(σ,τ)-Lie ideal of R such that τ commutes with ∗ and a,b be in R. (i) If a∈S_{∗}(R) and [U,a]=0 then a∈Z(R) or U⊂Z(R). (ii) If a∈S_{∗}(R) and [U,a]_{σ,τ}⊂ C_{σ,τ} then a∈Z(R) or U⊂Z(R). (iii) If U⊂Z(R) and U⊂C_{σ,τ} then there exists a nonzero ∗-ideal...

Let R be a *–prime ring with characteristic not 2; U be a nonzero *– (σ; τ)–Lie ideal of R and d be a nonzero derivation of R: Suppose σ, τ be two automorphisms of R such that σd = dσ; τd = dτ and * commutes with σ, τ, d: In the present paper it is shown that if d²(U) = (0); then U ⊆ Z.

In this paper, we study commutativity of a prime or semiprime ring using a map F : R 􀀀! R, multiplicative (generalized)-derivation and a map H : R 􀀀! R; multiplicative left centralizer, under the following condi- tions: For all x; y 2 R, i) F(xy) � H(xy) = 0; ii) F(xy) � H(yx) = 0; iii) F(x)F(y) � H(xy) = 0; iv) F(xy) � H(xy) 2 Z; v) F(xy) � H(yx)...

In this paper, we define a set including of all fa with a ∈ R generalized derivations of R and is denoted by f R. It is proved that (i) the mapping g : L (R) → f R given by g (a) = f −a for all a ∈ R is a Lie epimorphism with kernel Nσ,τ ; (ii) if R is a semiprime ring and σ is an epimorphism of R, the mapping h : f R → I (R) given by h (fa) = i σ(...

Let R be a semiprime ring and L be a semigroup ideal of R. The main object in this paper is to study

Let R be a prime ring, I be a nonzero semigroup ideal of R, d, g, h be derivations of R and a, b ∈ R. It is proved that if d(x) = ag(x)+h(x)b for all x ∈ I and a, b are not in Z(R) then there exists for some � ∈ C such that h(x) = � [a, x], g(x) = � [b, x] and d(x) = � [ab, x] for all x ∈ I.

Let R be a prime ring, I be a nonzero semigroup ideal of R, d, g, h be derivations of R and a, b ∈ R. It is proved that if d(x) = ag(x)+h(x)b for all x ∈ I and a, b are not in Z(R) then there exists for some λ ∈ C such that h(x) = λ [a, x], g(x) = λ [b, x] and d(x) = λ [ab, x] for all x ∈ I.

Let R be a prime ring, I be a nonzero semigroup ideal of R, d, g, h be derivations of R and a, b ∈ R. It is proved that if d(x) = ag(x)+h(x)b for all x ∈ I and a, b are not in Z(R) then there exists for some λ ∈ C such that h(x) = λ [a, x], g(x) = λ [b, x] and d(x) = λ [ab, x] for all x ∈ I.

IIn this paper, let X be a finite set, D be a complete X-semilattice of unions and Q = {T1, T2, T3, T4, T5, T6, T7, T8} be an X-subsemilattice of D where T1 T3 T5 T6 T8, T1 T3 T5 T7 T8, T2 T3 T5 T6 T8, T2 T3 T5 T7 ⊂ T8, T2 ⊂ T4 ⊂ T5 ⊂ T6 ⊂ T8, T2 ⊂ T4 ⊂ T5 ⊂ T7 ⊂ T8, T2\T1 ≠ φ, T1\T2 ≠ φ, T4\T3 ≠ φ, T3\T4≠ φ, T6\T7 ≠ φ, T7\T6 ≠ φ, T2 ⊂ T1 = T3, T4...

We investigate regular elements α and idempotents of the complete semigroup of binary relations B X (D) defined by semilattices of the class lower incomplete nets, for which V(D,α)=Q. Also we investigate right units of the semigroup B X (Q). For the case where X is a finite set we derive formulas by means of which we can calculate the numbers of re...

In this paper let Q = {T1, T2, T3, T 4, T5, T6, T7, T8} be a subsemilattice of X-semilattice of unions D where T1 ⊂ T 2 ⊂ T3 ⊂ T5 ⊂ T6 ⊂ T8, T1 ⊂ T2 ⊂ T3 ⊂ T5 ⊂ T7 ⊂ T8, T1 ⊂ T2 ⊂ T4 ⊂ T5 ⊂ T6 ⊂ T8, T1 ⊂ T2 ⊂ T4 ⊂ T5 ⊂ T7 ⊂ T8, T1 ≠ θ, T4\T3 ≠ θ, T3\T4 ≠ θ, T6\T7 ≠ θ, T7\T 6 ≠ θ, T3 ⊃ T4 = T5, T6 ⊃ T7 = T8, then we characterize the class each eleme...

Celiac disease or gluten sensitive enteropathy is an autoimmune disease characterized by inflammation of the small-bowel mucosa. As can be asymptomatic, involvement of the hematologic, gastrointestinal system, musculosceletal system, nervous system or endocrine system may occur as well. The presence of osteoporosis in celiac disease, may be the onl...

Common causes of bilateral lower extremity edema are venous insufficiency, pulmonary hypertension, heart failure, idiopathic edema, lymphedema, drugs (calcium channel blockers, beta blockers, clonidine, hormones, prednisone, and anti-inflammatory drugs), premenstrual edema, pregnancy and obesity. Among these, valproate-induced lower extremity edema...

Let N be a prime left near-ring with multiplicative center Z, and D be a (σ,τ)-derivation such that σD=Dσ and τD=Dτ. (i) If D(N)⊂Z, or [D(N),D(N)]=0 or [D(N),D(N)] σ,τ =0, then (N,+) is Abelian. (ii) If N is 2-torsion free, d 1 is a (σ,τ)-derivation and d 2 is a derivation on N such that d 1 d 2 (N)=0, then d 1 =0 or d 2 =0.

Let R be a 2-torsion free prime ring and let σ, τ be automorphisms of R. For any x, y ∈ R, set [x, y]σ, τ = xσ(y) - τ(y)x. Suppose that d is a (σ, τ)-derivation defined on R. In the present paper it is shown that (i) if d is a nonzero (σ, τ)-derivation and h is a nonzero derivation of R such that dh(R) ⊂ Cσ, τ then R is commutative, (ii) if R satis...

This paper abstracts some results of M. Bresar and J. Vukman [1] on the orthogonal derivations of semiprime rings to (σ, τ)-derivations and generalized (σ, τ)-derivations.

Let N be a 3-prime left near-ring with multiplicative center Z, a (σ, τ)-derivation D on N is defined to be an additive endomorphism satisfying the product rule D(xy) = τ(x)D(y) + D(x)σ(y) for all x, y ∈ N, where σ and τ are automorphisms of N. A nonempty subset U of N will be called a semigroup right ideal (resp. semigroup left ideal) if U N ⊂ U (...

The purpose of this paper is to show that every generalized Jordan derivation of prime ring with characteristic not two is a generalized derivation on a nonzero Lie ideal U of R such that which is a generalization of the well-known result of I. N. Herstein.

I. N. Herstein [Can. Math. Bull. 21, 369-370 (1978; Zbl 0412.16018)], has proved that if R is a prime ring and T is a Lie ideal of R such that [T,T]⊂Z, then T⊂Z. In the first part of this note, the above theorem is generalized for a (σ,τ)-left Lie ideal U of a prime ring. In the second part, some results are given for one-sided (σ,τ)-left Lie ideal...

Let R be a prime ring with characteristic not equal to two, σ, τ be automorphisms of R, and d be a nonzero derivation of R commuting with σ and τ. It is proved that for any (σ,τ)-left Lie ideal U of R: (1) if d(U)⊆Z, then σ(u)+τ(u)∈Z, for all u∈U, (2) if d 2 (U)=0, then σ(u)+τ(u)∈Z, for all u∈U, (3) if charR≠2,3, d(U)⊆U and d 2 (U)⊆Z, then σ(u)+τ(u...

Let R be a prime ring of characteristic not 2, U a nonzero ideal of R and 0≠da(α,β)-derivation of R where α and β are automorphisms of R. i) [d(U),a]=0 then a∈Z ii) For a,b∈R, the following conditions are equivalent (I) α(a)d(x)=d(x)β(b), for all x∈U (II) Either α(a)=β(b)∈CR(d(U)) or CR(a)=CR(b)=RâÂ...

In this paper, we proved some results for one-sided (σ, τ)-Lie ideals in prime rings.

Let R be a prime ring of characteristic not 2, U a nonzero ideal of R and 0≠da(α,β)-derivation of R where α and β are automorphisms of R. i) [d(U),a]=0 then a∈Z ii) For a,b∈R, the following conditions are equivalent (I) α(a)d(x)=d(x)β(b), for all x∈U (II) Either α(a)=β(b)∈CR(d(U)) or CR(a)=CR(b)=R′ and a[a,x]=[a,x]b (or a[b,x]=[b,x]b) for all x∈U....

Let R be a prime unit of charR≠2, σ,τ:R→R two automorphisms, U a nonzero (σ,τ)-Lie ideal of R and 0≠d:R→R a derivation such that σd=dσ, τd=dτ. (i) d(U)⊂C σ,τ then U⊂Z. (ii) If for a∈R, d(U)a=0 (or ad(U)=0) then a=0 or U⊂Z. (iii) If d 2 (U)=0 then U⊂Z.

Let R be a prime ring with involution, charR≠2 and σ and τ two automorphisms of R. Let d:R→R be a non-zero (σ,τ)-derivation and S the set of symmetric elements of R. In this case the following results are proved: (i) if d(S)⊂C σ,τ then S⊂Z; (ii) if a∈S and τ(a)d(S)=0 then a=0 or S⊂Z; (iii) if [d(S),S] σ,τ =0 and S⊄Z then S has no nilpotent element.

Let R be a prime ring of characteristic not 2, d:R→R a non-zero (σ,τ)-derivation, and U a non-zero ideal of R. In this case: (i) if a∈R and [d(U),a] σ,τ =0 then a∈Z, the center of R, (ii) if [d(U),d(U)] σ,τ =0 then R is commutative, (iii) if a∈R and [d(R),a] σ,τ ⊆C στ then a∈Z, (iv) if [d(R),d(R)] σ,τ ⊆C σ,τ then R is commutative.

Let N be a 3-prime left near-ring with multiplicative center Z, a (�, �)-derivation D on N is defined to be an additive endomorphism satisfying the product rule D(xy) = �(x)D(y)+D(x)�(y) for all x,y 2 N, whereand � are automorphisms of N. A nonempty subset U of N will be called a semigroup right ideal (resp. semigroup left ideal) if UNU (resp. NUU)...

Let N be a prime left near-ring with multiplicative centerZ; and D be a (α, γ)derivation such that δD = Dδ and ΓD = DΓ(i)If D(N)⊂ Z; or [D(N);D(N)] = 0 or [D(N);D(N)]σ, γ= 0; then (N; +)is abelian. (ii) If N is 2-torsion free, d1 is a (α, γ)-derivation and d2 is a derivation on N such that d1d2(N) = 0, then d1 = 0 or d2 = 0.