The study consists of two parts. The first part shows that if $h_{1}(x)h_{2}(y)=h_{3}(x)h_{4}(y)$... more The study consists of two parts. The first part shows that if $h_{1}(x)h_{2}(y)=h_{3}(x)h_{4}(y)$, for all $x,y\in R$, then $ h_{1}=h_{3}$ and $h_{2}=h_{4}$. Here, $h_{1},h_{2},h_{3},$ and $h_{4}$ are zero-power valued non-zero homoderivations of a prime ring $R$. Moreover, this study provide an explanation related to $h_{1}$ and $h_{2}$ satisfying the condition $ah_{1}+h_{2}b=0$. The second part shows that $L\subseteq Z$ if one of the following conditions is satisfied: $i. h(L)=(0)$, $ ii. h(L)\subseteq Z$, $iii. h(xy)=xy$, for all $x,y\in L$, $iv. h(xy)=yx$, for all $x,y\in L$, or $v. h([x,y])=0$, and for all $x,y\in L$. Here, $R$ is a prime ring with a characteristic other than $2$, $h$ is a homoderivation of $R$, and $L$ is a non-zero square closed Lie ideal of $R$.
Leonardo Fibonacci 13. yy yaşamış İtalyan bir matematikçidir. Fibonacci için “MatematiğiAraplar’d... more Leonardo Fibonacci 13. yy yaşamış İtalyan bir matematikçidir. Fibonacci için “MatematiğiAraplar’dan alıp, Avrupa’ya aktaran kişi” denilebilir. Fibonacci yazdığı Liber Abaci’ya adlıkitabında 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 oluşturulmuş gibi görünüyor olsa da bu sayılaradoğanın her yerinde rastlamak mümkündür. Örneğin, yavru bir salyangoz büyüdükçe kabuğunda yeniodacıklar oluşur. Her bir oda kendinden önceki iki odanın toplamı kadardır. Tıpkı Fibbonaccidizisindeki sayıların her birinin kendisinden önce gelen iki sayının toplamından oluşması gibi.Bir karaağacın, bir ıhlamur, erik, badem dallarındaki yaprakların ya da bir ayçiçeğindekitaneciklerin şaşırtıcı ve görkemli düzenini biliyor musunuz?Bir çiçeğin taç yapraklarının, soğan zarının, salatalık ya da çam yapraklarının düzeniylegösterdiği inanılmaz benzerliği hiç merak ediyor musunuz?Burada bir altın oran buluna...
Communications of The Korean Mathematical Society, 2017
In this paper, we define a set including of all fa with a ∈ R generalized derivations of R and is... more 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(−a) is a Lie epimorphism with kernel l (f R) ; (iii) if f R is a prime Lie ring and A, B are Lie ideals of R, then [f A , f B ] = (0) implies that either f A = (0) or f B = (0).
Bu calismada, asal halka uzerinde tanimli sifirdan farkli bir ters (𝛼,𝛽)− biturevin ayni zamanda ... more Bu calismada, asal halka uzerinde tanimli sifirdan farkli bir ters (𝛼,𝛽)− biturevin ayni zamanda (𝛼,𝛽)− biturev oldugu ispatlanmistir. Ayrica, 𝑐ℎ𝑎𝑟(𝑅)≠2 olacak bicimdeki degismeli olmayan bir yari-asal 𝑅 halkasi uzerinde tanimli Jordan (𝛼,𝛽)− biturevin ayni zamanda (𝛼,𝛽)− biturev oldugu gosterilmistir. Bunlarin yaninda, Jordan sol (𝛼,𝛼)−biturevli asal halkalarin degismeli olma ozellikleri arastirilmistir.
Let N be a prime left near-ring with multiplicative centerZ; and D be a (α, γ)derivatio... more 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.</p
In this study, we prove that any nonzero reverse (,) − biderivation on a prime ring is (,) − bide... more In this study, we prove that any nonzero reverse (,) − biderivation on a prime ring is (,) − biderivation. Also, we show that any Jordan (,) − biderivation on non-commutative semi-prime ring with ℎ () ≠ 2 is an (,) − biderivation. In addition, we investigate commutative feature of prime ring with Jordan left (,) − biderivation.
The algebraic properties and identities of a semiprime ring are investigated with the help of the... more 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.
In this paper, we take Q = { T1, T 2, . . . , Tm−3, Tm−2, Tm−1, Tm } subsemilattice of X−semilatt... more In this paper, we take Q = { T1, T 2, . . . , Tm−3, Tm−2, Tm−1, Tm } subsemilattice of X−semilattice of unions D where the elements Ti’ s are satisfying the following properties, T1⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−2⊂ Tm, T1⊂ T 3⊂ · · · ⊂ Tm−3 ⊂ Tm−1⊂ Tm, T2⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−2⊂ Tm, T2⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−1 ⊂ Tm, T1\T 2 6= ∅, T2\T 1 6= ∅, Tm−2\Tm−1 6= ∅, Tm−1\Tm−2 6= ∅, T1∪T 2= T 3, Tm−2∪Tm−1= Tm. We will investigate the properties of regular element α ∈ BX(D) satisfying V (D,α) = Q. Moreover, we will calculate the number of regular elements of BX(D) for a finite set X.
Journal of Mathematical and Computational Science, 2017
Let R be a semiprime ring and L be a semigroup ideal of R. The main object in this paper is to st... more Let R be a semiprime ring and L be a semigroup ideal of R. The main object in this paper is to study the following situations in semiprime rings: When F is a multiplicative (α,1)-(generalized) derivation associated with a map d, (i) F(xy)±α(x)α(y)=0 for x,y∈L. (ii) F(x)F(y)±α(x)α(y)=0 for all x,y∈L. When F is a multiplicative (1,α)-(generalized) derivation associated with a map d, (iii) F(xy)±xy=0 for all x,y∈L. (iv) F(x)F(y)±xy=0 for all x,y∈L.
Let R be a σ-prime ring with characteristic not 2, Z(R) be the center of R, I be a nonzero σ-idea... more Let R be a σ-prime ring with characteristic not 2, Z(R) be the center of R, I be a nonzero σ-ideal of R, α, β : R→ R be two automorphisms, d be a nonzero (α, β)-derivation of R and h be a nonzero derivation of R. In the present paper, it is shown that (i) If d (I) ⊂ Cα,β and β commutes with σ then R is commutative. (ii) Let α and β commute with σ. If a ∈ I ∩ Sσ (R) and [d(I), a]α,β ⊂ Cα,β then a ∈ Z(R). (iii) Let α, β and h commute with σ. If dh (I) ⊂ Cα,β and h (I) ⊂ I then R is commutative.
The study consists of two parts. The first part shows that if $h_{1}(x)h_{2}(y)=h_{3}(x)h_{4}(y)$... more The study consists of two parts. The first part shows that if $h_{1}(x)h_{2}(y)=h_{3}(x)h_{4}(y)$, for all $x,y\in R$, then $ h_{1}=h_{3}$ and $h_{2}=h_{4}$. Here, $h_{1},h_{2},h_{3},$ and $h_{4}$ are zero-power valued non-zero homoderivations of a prime ring $R$. Moreover, this study provide an explanation related to $h_{1}$ and $h_{2}$ satisfying the condition $ah_{1}+h_{2}b=0$. The second part shows that $L\subseteq Z$ if one of the following conditions is satisfied: $i. h(L)=(0)$, $ ii. h(L)\subseteq Z$, $iii. h(xy)=xy$, for all $x,y\in L$, $iv. h(xy)=yx$, for all $x,y\in L$, or $v. h([x,y])=0$, and for all $x,y\in L$. Here, $R$ is a prime ring with a characteristic other than $2$, $h$ is a homoderivation of $R$, and $L$ is a non-zero square closed Lie ideal of $R$.
Leonardo Fibonacci 13. yy yaşamış İtalyan bir matematikçidir. Fibonacci için “MatematiğiAraplar’d... more Leonardo Fibonacci 13. yy yaşamış İtalyan bir matematikçidir. Fibonacci için “MatematiğiAraplar’dan alıp, Avrupa’ya aktaran kişi” denilebilir. Fibonacci yazdığı Liber Abaci’ya adlıkitabında 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 oluşturulmuş gibi görünüyor olsa da bu sayılaradoğanın her yerinde rastlamak mümkündür. Örneğin, yavru bir salyangoz büyüdükçe kabuğunda yeniodacıklar oluşur. Her bir oda kendinden önceki iki odanın toplamı kadardır. Tıpkı Fibbonaccidizisindeki sayıların her birinin kendisinden önce gelen iki sayının toplamından oluşması gibi.Bir karaağacın, bir ıhlamur, erik, badem dallarındaki yaprakların ya da bir ayçiçeğindekitaneciklerin şaşırtıcı ve görkemli düzenini biliyor musunuz?Bir çiçeğin taç yapraklarının, soğan zarının, salatalık ya da çam yapraklarının düzeniylegösterdiği inanılmaz benzerliği hiç merak ediyor musunuz?Burada bir altın oran buluna...
Communications of The Korean Mathematical Society, 2017
In this paper, we define a set including of all fa with a ∈ R generalized derivations of R and is... more 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(−a) is a Lie epimorphism with kernel l (f R) ; (iii) if f R is a prime Lie ring and A, B are Lie ideals of R, then [f A , f B ] = (0) implies that either f A = (0) or f B = (0).
Bu calismada, asal halka uzerinde tanimli sifirdan farkli bir ters (𝛼,𝛽)− biturevin ayni zamanda ... more Bu calismada, asal halka uzerinde tanimli sifirdan farkli bir ters (𝛼,𝛽)− biturevin ayni zamanda (𝛼,𝛽)− biturev oldugu ispatlanmistir. Ayrica, 𝑐ℎ𝑎𝑟(𝑅)≠2 olacak bicimdeki degismeli olmayan bir yari-asal 𝑅 halkasi uzerinde tanimli Jordan (𝛼,𝛽)− biturevin ayni zamanda (𝛼,𝛽)− biturev oldugu gosterilmistir. Bunlarin yaninda, Jordan sol (𝛼,𝛼)−biturevli asal halkalarin degismeli olma ozellikleri arastirilmistir.
Let N be a prime left near-ring with multiplicative centerZ; and D be a (α, γ)derivatio... more 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.</p
In this study, we prove that any nonzero reverse (,) − biderivation on a prime ring is (,) − bide... more In this study, we prove that any nonzero reverse (,) − biderivation on a prime ring is (,) − biderivation. Also, we show that any Jordan (,) − biderivation on non-commutative semi-prime ring with ℎ () ≠ 2 is an (,) − biderivation. In addition, we investigate commutative feature of prime ring with Jordan left (,) − biderivation.
The algebraic properties and identities of a semiprime ring are investigated with the help of the... more 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.
In this paper, we take Q = { T1, T 2, . . . , Tm−3, Tm−2, Tm−1, Tm } subsemilattice of X−semilatt... more In this paper, we take Q = { T1, T 2, . . . , Tm−3, Tm−2, Tm−1, Tm } subsemilattice of X−semilattice of unions D where the elements Ti’ s are satisfying the following properties, T1⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−2⊂ Tm, T1⊂ T 3⊂ · · · ⊂ Tm−3 ⊂ Tm−1⊂ Tm, T2⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−2⊂ Tm, T2⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−1 ⊂ Tm, T1\T 2 6= ∅, T2\T 1 6= ∅, Tm−2\Tm−1 6= ∅, Tm−1\Tm−2 6= ∅, T1∪T 2= T 3, Tm−2∪Tm−1= Tm. We will investigate the properties of regular element α ∈ BX(D) satisfying V (D,α) = Q. Moreover, we will calculate the number of regular elements of BX(D) for a finite set X.
Journal of Mathematical and Computational Science, 2017
Let R be a semiprime ring and L be a semigroup ideal of R. The main object in this paper is to st... more Let R be a semiprime ring and L be a semigroup ideal of R. The main object in this paper is to study the following situations in semiprime rings: When F is a multiplicative (α,1)-(generalized) derivation associated with a map d, (i) F(xy)±α(x)α(y)=0 for x,y∈L. (ii) F(x)F(y)±α(x)α(y)=0 for all x,y∈L. When F is a multiplicative (1,α)-(generalized) derivation associated with a map d, (iii) F(xy)±xy=0 for all x,y∈L. (iv) F(x)F(y)±xy=0 for all x,y∈L.
Let R be a σ-prime ring with characteristic not 2, Z(R) be the center of R, I be a nonzero σ-idea... more Let R be a σ-prime ring with characteristic not 2, Z(R) be the center of R, I be a nonzero σ-ideal of R, α, β : R→ R be two automorphisms, d be a nonzero (α, β)-derivation of R and h be a nonzero derivation of R. In the present paper, it is shown that (i) If d (I) ⊂ Cα,β and β commutes with σ then R is commutative. (ii) Let α and β commute with σ. If a ∈ I ∩ Sσ (R) and [d(I), a]α,β ⊂ Cα,β then a ∈ Z(R). (iii) Let α, β and h commute with σ. If dh (I) ⊂ Cα,β and h (I) ⊂ I then R is commutative.
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