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Published: 6 September 2023

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1

Chakraborty, Sujoy, and Akhil Chandra Paul. "Jordan k-derivations of certain Nobuswa gamma rings." GANIT: Journal of Bangladesh Mathematical Society 31 (April 9, 2012): 53–64. http://dx.doi.org/10.3329/ganit.v31i0.10308.

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From the very definition, it follows that every Jordan k-derivation of a gamma ring M is, in general, not a k-derivation of M. In this article, we establish its generalization by considering M as a 2-torsion free semiprime ?N-ring (Nobusawa gamma ring). We also show that every Jordan k-derivation of a 2-torsion free completely semiprime ?N-ring is a k-derivation of the same.DOI: http://dx.doi.org/10.3329/ganit.v31i0.10308GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 53-64

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2

Ashraf, Mohammad, Abdelkarim Boua, and Mohammad Aslam Siddeeque. "Generalized multiplicative derivations in 3-prime near rings." Mathematica Slovaca 68, no. 2 (April 25, 2018): 331–38. http://dx.doi.org/10.1515/ms-2017-0104.

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Abstract In the present paper, we introduce the notion of generalized multiplicative derivation in a near-ring N and investigate commutativity of 3-prime near-rings, showing that certain conditions involving generalized multiplicative derivations force N to be a commutative ring. Finally some more results related with the structure of these derivations are also obtained.

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3

Ahmed, Wasim, Muzibur Rahman Mozumder, and Arshad Madni. "Study of multiplicative derivation and its additivity." Proyecciones (Antofagasta) 42, no. 1 (February 1, 2023): 219–32. http://dx.doi.org/10.22199/issn.0717-6279-5578.

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In this paper, we modify the result of M. N. Daif [1] on multiplicative derivations in rings. He showed that the multiplicative derivation is additive by imposing certain conditions on the ring ℜ. Here, we have proved the above result with lesser conditions than M. N. Daif for getting multiplicative derivation to be additive.

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4

Dey, K. K., and A. C. Paul. "Permuting Tri-Derivations of Semiprime Gamma Rings." Journal of Scientific Research 5, no. 1 (December 26, 2012): 55–66. http://dx.doi.org/10.3329/jsr.v5i1.9549.

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We study some properties of permuting tri-derivations on semiprime G-rings with a certain assumption. Let M be a 3-torsion free semiprime G-ring satisfying a certain assumption and let I be a non-zero ideal of M. Suppose that there exists a permuting tri-derivation D: M×M´M ? M such that d is an automorphism commuting on I and also d is a trace of D. Then we prove that I is a nonzero commutative ideal. Various characterizations of M are obtained by means of tri-derivations.© 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi: http://dx.doi.org/10.3329/jsr.v5i1.9549 J. Sci. Res. 5 (1), 55-66 (2013)

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5

Ashraf, Mohammad, and Mohammad Aslam Siddeeque. "ON (σ, τ)-n-DERIVATIONS IN NEAR-RINGS." Asian-European Journal of Mathematics 06, no. 04 (December 2013): 1350051. http://dx.doi.org/10.1142/s1793557113500514.

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In the present paper, we introduce the notion of (σ, τ)-n-derivation in near-ring N and investigate some properties involving (σ, τ)-n-derivations of a prime near-ring N which force N to be a commutative ring. Additive commutativity of near-ring N satisfying certain identities involving (σ, τ)-n-derivations of a prime near-ring N has also been obtained. Related examples to justify the hypotheses in various theorems have also been provided.

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6

BOUA, ABDELKARIM. "Some Identities in Rings and Near-Rings with Derivations." Kragujevac Journal of Mathematics 45, no. 01 (March 2021): 75–80. http://dx.doi.org/10.46793/kgjmat2101.075b.

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In the present paper we investigate commutativity in prime rings and 3-prime near-rings admitting a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings and 3-prime near-rings have been generalized.

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7

Širovnik, Nejc, and Joso Vukman. "On Certain Functional Equation in Semiprime Rings." Algebra Colloquium 23, no. 01 (January 6, 2016): 65–70. http://dx.doi.org/10.1142/s1005386716000080.

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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 [Formula: see text] for all x ∈ 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 ∈ R is some fixed element.

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8

Dey, Kalyan Kumar, and Akhil Chandra Paul. "Semi derivations of prime gamma rings." GANIT: Journal of Bangladesh Mathematical Society 31 (April 9, 2012): 65–70. http://dx.doi.org/10.3329/ganit.v31i0.10309.

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Let M be a prime ?-ring satisfying a certain assumption (*). An additive mapping f : M ? M is a semi-derivation if f(x?y) = f(x)?g(y) + x?f(y) = f(x)?y + g(x)?f(y) and f(g(x)) = g(f(x)) for all x, y?M and ? ? ?, where g : M?M is an associated function. In this paper, we generalize some properties of prime rings with semi-derivations to the prime &Gamma-rings with semi-derivations. 2000 AMS Subject Classifications: 16A70, 16A72, 16A10.DOI: http://dx.doi.org/10.3329/ganit.v31i0.10309GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 65-70

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9

Boua, A., L. Oukhtite, and A. Raji. "On generalized semiderivations in 3-prime near-rings." Asian-European Journal of Mathematics 09, no. 02 (April 15, 2016): 1650036. http://dx.doi.org/10.1142/s1793557116500364.

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There is a large body of evidence showing that the existence of a suitably-constrained derivation on a 3-prime near-ring forces the near-ring to be a commutative ring. The purpose of this paper is to study generalized semiderivations which satisfy certain identities on 3-prime near-ring and generalize some results due to [H. E. Bell and G. Mason, On derivations in near-rings, North-Holland Math. Stud. 137 (1987) 31–35; H. E. Bell, On prime near-rings with generalized derivation, Int. J. Math. Math. Sci. 2008 (2008), Article ID: 490316, 5[Formula: see text]pp; A. Boua and L. Oukhtite, Some conditions under which near-rings are rings, Southeast Asian Bull. Math. 37 (2013) 325–331]. Moreover, an example is given to prove that the necessity of the 3-primeness hypothesis imposed on the various theorems cannot be marginalized.

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10

Liang, Xinfeng, and Lingling Zhao. "Bi-Lie n-derivations on triangular rings." AIMS Mathematics 8, no. 7 (2023): 15411–26. http://dx.doi.org/10.3934/math.2023787.

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<abstract><p>The purpose of this article is to prove that every bi-Lie n-derivation of certain triangular rings is the sum of an inner biderivation, an extremal biderivation and an additive central mapping vanishing at $ (n-1)^{th} $-commutators for both components, using the notion of maximal left ring of quotients. As a consequence, we characterize the decomposition structure of bi-Lie n-derivations on upper triangular matrix rings.</p></abstract>

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11

Brešar, M., and J. Vukman. "On certain subrings of prime rings with derivations." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 54, no. 1 (February 1993): 133–41. http://dx.doi.org/10.1017/s1446788700037046.

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AbstractLet D be a nonzero derivation of a noncommutative prime ring R, and let U be the subring of R generated by all [D(x), x], x ∞ R. A classical theorem of Posner asserts that U is not contained in the center of R. Under the additional assumption that the characteristic of R is not 2, we prove a more general result stating that U contains a nonzero left ideal of R as well as a nonzero right ideal of R.

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12

Kikumasa, Isao. "Automorphisms of a Certain Skew Polynomial Ring of Derivation Type." Canadian Journal of Mathematics 42, no. 6 (December 1, 1990): 949–58. http://dx.doi.org/10.4153/cjm-1990-050-0.

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Throughout this paper, all rings have the identity 1 and ring homomorphisms are assumed to preserve 1. We use p to denote a prime integer and F to denote a field of characteristic p. For an element α in F, we setA = F[ϰ]/(ϰp - α)F[ϰ].Moreover, by D and R, we denote the derivation of A induced by the ordinary derivation of F[ϰ] and the skew polynomial ring A[X,D] where aX = Xa+D(a) (a ∈ A), respectively (cf. [2]).In [3], R. W. Gilmer determined all the B-automorphisms of B[X] for any commutative ring B. Since then, some extensions or generalizations of his results have been obtained ([1], [2] and [5]). As to the characterization of automorphisms of skew polynomial rings, M. Rimmer [5] established a thorough result in case of automorphism type, while M. Ferrero and K. Kishimoto [2], among others, have made some progress in case of derivation type.

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13

Abu Nawas, Mohammad Khalil, and Radwan M. Al-Omary. "On Ideals and Commutativity of Prime Rings with Generalized Derivations." European Journal of Pure and Applied Mathematics 11, no. 1 (January 30, 2018): 79. http://dx.doi.org/10.29020/nybg.ejpam.v11i1.3142.

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An additive mapping F: R → R is called a generalized derivation on R if there exists a derivation d: R → R such that F(xy) = xF(y) + d(x)y holds for all x,y ∈ R. It is called a generalized (α,β)−derivation on R if there exists an (α,β)−derivation d: R → R such that the equation F(xy) = F(x)α(y)+β(x)d(y) holds for all x,y ∈ R. In the present paper, we investigate commutativity of a prime ring R, which satisﬁes certain diﬀerential identities on left ideals of R. Moreover some results on commutativity of rings with involutions that satisfy certain identities are proved.

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14

Fošner, Maja, Benjamin Marcen, and Joso Vukman. "On certain functional equation in prime rings." Open Mathematics 20, no. 1 (January 1, 2022): 140–52. http://dx.doi.org/10.1515/math-2022-0002.

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Abstract The purpose of this paper is to prove the following result. Let R R be prime ring of characteristic different from two and three, and let F : R → R F:R\to R be an additive mapping satisfying the relation F ( x 3 ) = F ( x 2 ) x − x F ( x ) x + x F ( x 2 ) F\left({x}^{3})=F\left({x}^{2})x-xF\left(x)x+xF\left({x}^{2}) for all x ∈ R x\in R . In this case, F F is of the form 4 F ( x ) = D ( x ) + q x + x q 4F\left(x)=D\left(x)+qx+xq for all x ∈ R x\in R , where D : R → R D:R\to R is a derivation, and q q is some fixed element from the symmetric Martindale ring of quotients of R R .

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15

Atteya, Mehsin Jabel. "Notes on the higher derivations on prime rings." Boletim da Sociedade Paranaense de Matemática 37, no. 4 (January 9, 2018): 61–68. http://dx.doi.org/10.5269/bspm.v37i4.32357.

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The main purpose of thess notes investigated some certain properties and relation between higher derivation (HD,for short) and Lie ideal of semiprime rings and prime rings,we gave some results about that.

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16

Ashraf, Mohammad, and Bilal Ahmad Wani. "On certain functional equations related to Jordan *-derivations in semiprime *-rings and standard operator algebras." Pure Mathematics and Applications 27, no. 1 (July 1, 2018): 1–17. http://dx.doi.org/10.1515/puma-2015-0024.

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Abstract The purpose of this paper is to investigate identities with Jordan *-derivations in semiprime *-rings. Let ℛ be a 2-torsion free semiprime *-ring. In this paper it has been shown that, if ℛ admits an additive mapping D : ℛ→ℛsatisfying either D(xyx) = D(xy)x*+ xyD(x) for all x,y ∈ ℛ, or D(xyx) = D(x)y*x*+ xD(yx) for all pairs x, y ∈ ℛ, then D is a *-derivation. Moreover this result makes it possible to prove that if ℛ satis es 2D(xn) = D(xn−1)x* + xn−1D(x) + D(x)(x*)n−1 + xD(xn−1) for all x ∈ ℛ and some xed integer n ≥ 2, then D is a Jordan *-derivation under some torsion restrictions. Finally, we apply these purely ring theoretic results to standard operator algebras 𝒜(ℋ). In particular, we prove that if ℋ be a real or complex Hilbert space, with dim(ℋ) > 1, admitting a linear mapping D : 𝒜(ℋ) → ℬ(ℋ) (where ℬ(ℋ) stands for the bounded linear operators) such that $$2D\left( {A^n } \right) = D\left( {A^{n - 1} } \right)A^* + A^{n - 1} D\left( A \right) + D\left( A \right)\left( {A^* } \right)^{n - 1} + AD\left( {A^{n - 1} } \right)$$ for all A∈𝒜(ℋ). Then D is of the form D(A) = AB−BA* for all A∈𝒜(ℋ) and some fixed B ∈ ℬ(ℋ), which means that D is Jordan *-derivation.

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17

Nielsen, Pace P., and Michał Ziembowski. "Derivations and bounded nilpotence index." International Journal of Algebra and Computation 25, no. 03 (April 9, 2015): 433–38. http://dx.doi.org/10.1142/s0218196715500034.

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We construct a nil ring R which has bounded index of nilpotence 2, is Wedderburn radical, and is commutative, and which also has a derivation δ for which the differential polynomial ring R[x;δ] is not even prime radical. This example gives a strong barrier to lifting certain radical properties from rings to differential polynomial rings. It also demarcates the strength of recent results about locally nilpotent PI rings.

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18

Dey, K. K., and A. C. Paul. "Commutativity of Prime Gamma Rings with Left Centralizers." Journal of Scientific Research 6, no. 1 (December 26, 2013): 69–77. http://dx.doi.org/10.3329/jsr.v6i1.14872.

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Let M be a G-ring. If M satisfies the condition (*) xaybz = xbyaz for all x, y, zÎM, a, bÎG, then we investigate commutativity of prime G-rings satisfying certain identities involving left centralizer. Keywords: Prime G-ring; Derivation; Generalized derivation; Left centralizer. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v6i1.14872 J. Sci. Res. 6 (1), 69-77 (2014)

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19

Ashraf, Mohammad, and Nazia Parveen. "Jordan Higher Derivable Mappings on Rings." Algebra 2014 (November 19, 2014): 1–9. http://dx.doi.org/10.1155/2014/672387.

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Let R be a ring. We say that a family of maps D={dn}n∈N is a Jordan higher derivable map (without assumption of additivity) on R if d0=IR (the identity map on R) and dn(ab+ba)=∑p+q=n‍dp(a)dq(b)+∑p+q=n‍dp(b)dq(a) hold for all a,b∈R and for each n∈N. In this paper, we show that every Jordan higher derivable map on a ring under certain assumptions becomes a higher derivation. As its application, we get that every Jordan higher derivable map on Banach algebra is an additive higher derivation.

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20

Saed, Ikram A. "Commutativity of Addition in Prime Near-Rings with Right (Î¸,Î¸)-3-Derivations." JOURNAL OF ADVANCES IN MATHEMATICS 14, no. 1 (April 13, 2018): 7533–39. http://dx.doi.org/10.24297/jam.v14i1.7185.

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Let N be a near-ring and is a mapping on N . In this paper we introduce the notion of right ()-3-derivation in near-ring N. Also, we investigate the commutativity of addition of prime near-rings satisfying certain identities involving right ()-3-derivation.

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A. Saed, Ikram. "SOME RESULTS OF GENERALIZED LEFT (Î¸,Î¸)-DERIVATIONS ON SEMIPRIME RINGS." JOURNAL OF ADVANCES IN MATHEMATICS 11, no. 8 (December 10, 2015): 5529–35. http://dx.doi.org/10.24297/jam.v11i8.1207.

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Let R be an associative ring with center Z(R) . In this paper , we study the commutativity of semiprime rings under certain conditions , it comes through introduce the definition of generalized left(Î¸,Î¸)- derivation associated with left (Î¸,Î¸) -derivation , where Î¸ is a mapping on R .

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22

Boua, Abdelkarim, and Mohammed Ashraf. "Generalized semiderivations in prime rings with algebraic identities." MATHEMATICA 62 (85), no. 2 (November 15, 2020): 148–59. http://dx.doi.org/10.24193/mathcluj.2020.2.04.

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Let R be a prime ring with center Z(R). Suppose that R admits a generalized semiderivation F with non-zero associated derivation d. We investigate the commutativity of a prime ring R satisfying certain identities.

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23

Zemzami, Omar Ait, Lahcen Oukhtite, Shakir Ali, and Najat Muthana. "On certain classes of generalized derivations." Mathematica Slovaca 69, no. 5 (October 25, 2019): 1023–32. http://dx.doi.org/10.1515/ms-2017-0286.

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Abstract Our purpose in this paper is to investigate some particular classes of generalized derivations and their relationship with commutativity of prime rings with involution. Some well-known results characterizing commutativity of prime rings have been generalized. Furthermore, we provide examples to show that the assumed restrictions cannot be relaxed.

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Dey, Kalyan Kumar, and Akhil Chandra Paul. "Generalized Derivations of Prime Gamma Rings." GANIT: Journal of Bangladesh Mathematical Society 33 (January 13, 2014): 33–39. http://dx.doi.org/10.3329/ganit.v33i0.17654.

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Let M be a prime ?-ring satisfying a certain assumption a?b?c = a?b?c for all a, b, c?M and ?, ???, and let I be an ideal of M. Assume that (D, d) is a generalized derivation of M and a?M. If D([x, a]?) = 0 or [D(x), a]? = 0 for all x?I, ? ? ?, then we prove that d(x) = p?[x, a]? for all x?I, ?, ? ? ? or a?Z(M) (the centre of M), where p belongs C(M) (the extended centroid of M). GANIT J. Bangladesh Math. Soc. Vol. 33 (2013) 33-39 DOI: http://dx.doi.org/10.3329/ganit.v33i0.17654

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25

Idrissi, My Abdallah, and Lahcen Oukhtite. "Some commutativity theorems for rings with involution involving generalized derivations." Asian-European Journal of Mathematics 12, no. 01 (February 2019): 1950001. http://dx.doi.org/10.1142/s1793557119500013.

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Our purpose in this paper is to investigate commutativity of a ring with involution [Formula: see text] which admits a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

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26

Atteya, Mehsin, Dalal Resan, and Mesaa Salmaan. "Certain Pairs of Generalized Derivations of Semiprime Rings." Journal of Al-Rafidain University College For Sciences ( Print ISSN: 1681-6870 ,Online ISSN: 2790-2293 ), no. 2 (October 17, 2021): 247–64. http://dx.doi.org/10.55562/jrucs.v32i2.338.

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Let R be a semiprime ring with the cancellation property ,U be a set of R, (D,d) and (G,g) be generalized derivations of R ,if R admits to satisfied some conditions ,where d acts as a left centralizer (resp.g acts as a left centralizer).Then there exist idempotents ϵ1, ϵ 2, ϵ 3 C and an invertible element λ C such that ϵ iϵ j= 0, for i≠j, ϵ 1+ ϵ 2+ ϵ 3 =1,and ϵ 1f(x)=λϵ 1g(x), ϵ 2g(x)= 0, ϵ 3f(x)= 0 hold for all x U. (*)During our paper we will using (*)for denoted to above result.

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27

Wong, Tsai-Lien. "On Certain Subgroups of Semiprime Rings with Derivations." Communications in Algebra 32, no. 5 (December 31, 2004): 1961–68. http://dx.doi.org/10.1081/agb-120029916.

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Chebotar, M. A., and Pjek-Hwee Lee. "ON CERTAIN SUBGROUPS OF PRIME RINGS WITH DERIVATIONS." Communications in Algebra 29, no. 7 (May 31, 2001): 3083–87. http://dx.doi.org/10.1081/agb-5008.

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29

Bell, H. E., and L. C. Kappe. "Rings in which derivations satisfy certain algebraic conditions." Acta Mathematica Hungarica 53, no. 3-4 (September 1989): 339–46. http://dx.doi.org/10.1007/bf01953371.

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Mamouni, Abdellah, Lahcen Oukhtite, and Mohammed Zerra. "Some results on derivations and generalized derivations in rings." MATHEMATICA 65 (88), no. 1 (June 15, 2023): 94–104. http://dx.doi.org/10.24193/mathcluj.2023.1.10.

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The purpose of this paper is to study derivations and generalized derivations in prime rings satisfying certain differential identities. Some well-known results characterizing commutativity of prime rings have been generalized. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

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31

Fošner, A., N. Baydar, and R. Strašek. "Remarks on Certain Identities with Derivations on Semiprime Rings." Ukrainian Mathematical Journal 66, no. 10 (March 2015): 1609–14. http://dx.doi.org/10.1007/s11253-015-1037-9.

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32

Chen, Zhengxin, and Dengyin Wang. "Derivations of Certain Nilpotent Lie Algebras Over Commutative Rings." Communications in Algebra 39, no. 10 (October 2011): 3736–52. http://dx.doi.org/10.1080/00927872.2010.512581.

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33

Dadhwal, Madhu, and Geeta Devi. "On Lie ideals and multiplicative generalized (α, β)-reverse derivations of *-prime rings." Gulf Journal of Mathematics 14, no. 2 (April 4, 2023): 16–29. http://dx.doi.org/10.56947/gjom.v14i2.1000.

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In this paper, the notion of multiplicative generalized (α, β)-reverse derivations associated with (α, β)-reverse derivations of *-prime rings is characterized. The action of these derivations on *-Lie ideals of *-prime rings is also investigated. Moreover, the commutativity of *-prime rings admitting multiplicative generalized (α, β)-reverse derivations associated with (α, β)-reverse derivations satisfying certain algebraic identities on *-Lie ideals is explored.

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Kamal, Ahmed A. M. "$ \sigma $-derivations on prime near-rings." Tamkang Journal of Mathematics 32, no. 2 (June 30, 2001): 89–93. http://dx.doi.org/10.5556/j.tkjm.32.2001.349.

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The literatrue on near-rings contains a number of theorems asserting that certain conditions implying commutativity in rings imply multiplicative or additive commutativity in special classes of near-rings. H. E. Bell and G. Mason in [2] added to this body of results several commutativity theorems for near-rings admitting suitably-constrained derivations. In this paper we generalize some of their results to a subclass of prime near-rings admitting suitably-constrained $ \sigma $-derivations, where $ \sigma $ is an automorphibm of the prime near-ring.

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Dhara, B., S. Ghosh, and G. S. Sandhu. "On Lie ideals satisfying certain differential identities in prime rings." Extracta Mathematicae 38, no. 1 (June 1, 2023): 67–84. http://dx.doi.org/10.17398/2605-5686.38.1.67.

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Let R be a prime ring of characteristic not 2, L a nonzero square closed Lie ideal of R and let F : R → R, G : R → R be generalized derivations associated with derivations d : R → R, g : R → R respectively. In this paper, we study several conditions that imply that the Lie ideal is central. Moreover, it is shown that the assumption of primeness of R can not be removed.

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36

COJUHARI, E. P., and B. J. GARDNER. "GENERALIZED HIGHER DERIVATIONS." Bulletin of the Australian Mathematical Society 86, no. 2 (January 6, 2012): 266–81. http://dx.doi.org/10.1017/s000497271100308x.

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AbstractA type of generalized higher derivation consisting of a collection of self-mappings of a ring associated with a monoid, and here called a D-structure, is studied. Such structures were previously used to define various kinds of ‘skew’ or ‘twisted’ monoid rings. We show how certain gradings by monoids define D-structures. The monoid ring defined by such a structure corresponding to a group-grading is the variant of the group ring introduced by Năstăsescu, while in the case of a cyclic group of order two, the form of the D-structure itself yields some gradability criteria of Bakhturin and Parmenter. A partial description is obtained of the D-structures associated with infinite cyclic monoids.

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37

El Hamdaoui, Mouhamadi, Abdelkarim Boua, and Gurninder S. Sandhu. "Some identities in quotient rings." Boletim da Sociedade Paranaense de Matemática 41 (December 27, 2022): 1–9. http://dx.doi.org/10.5269/bspm.62481.

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Let R be an associative ring, P a prime ideal of R: In this paper, we study the structure of the ring R=P and describe the possible forms of the generalized derivations satisfying certain algebraic identities on R: As a consequence of our theorems, we first investigate strong commutativity preserving generalized derivations of prime rings, and then examine the generalized derivations acting as (anti)homomorphisms in prime rings. Some commutativity theorems also given in semi-prime rings.

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38

Boua, Abdelkarim. "Commutativity of near-rings with certain constrains on Jordan ideals." Boletim da Sociedade Paranaense de Matemática 36, no. 4 (October 1, 2018): 159–70. http://dx.doi.org/10.5269/bspm.v36i4.32032.

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The purpose of this paper is to study derivations satisfying certain differential identities on Jordan ideals of 3-prime near-rings. Moreover, we provide examples to show that hypothesis of our results are necessary.

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39

Chuang, Chen-Lian, and Tsiu-Kwen Lee. "A NOTE ON CERTAIN SUBGROUPS OF PRIME RINGS WITH DERIVATIONS." Communications in Algebra 30, no. 7 (August 7, 2002): 3259–65. http://dx.doi.org/10.1081/agb-120004486.

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40

Nejjar, Badr, Ali Kacha, Abdellah Mamouni, and Lahcen Oukhtite. "Certain commutativity criteria for rings with involution involving generalized derivations." Georgian Mathematical Journal 27, no. 1 (March 1, 2020): 133–39. http://dx.doi.org/10.1515/gmj-2018-0010.

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AbstractIn this article we investigate some commutativity criteria for a ring with involution {(R,\ast}) in which generalized derivations satisfy certain algebraic identities. Moreover, we provide examples to show that the assumed restriction cannot be relaxed.

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41

Traves, William N. "Localization of the Hasse-Schmidt Algebra." Canadian Mathematical Bulletin 46, no. 2 (June 1, 2003): 304–9. http://dx.doi.org/10.4153/cmb-2003-031-2.

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AbstractThe behaviour of theHasse-Schmidt algebra of higher derivations under localization is studied using André cohomology. Elementary techniques are used to describe the Hasse-Schmidt derivations on certain monomial rings in the nonmodular case. The localization conjecture is then verified for all monomial rings.

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42

Mamouni, A., L. Oukhtite, and M. Samman. "Commutativity Theorems for *-Prime Rings with Differential Identities on Jordan Ideals." ISRN Algebra 2012 (December 29, 2012): 1–11. http://dx.doi.org/10.5402/2012/729356.

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In this paper we explore commutativity of -prime rings in which derivations satisfy certain differential identities on Jordan ideals. Furthermore, examples are given to demonstrate that our results cannot be extended to semiprime rings.

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43

Chen, Zhengxin. "Automorphisms and Derivations of Certain Solvable Lie Algebras Over Commutative Rings." Communications in Algebra 40, no. 2 (February 2012): 738–69. http://dx.doi.org/10.1080/00927872.2010.536961.

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44

Paykan, Kamal, and Ahmad Moussavi. "Study of skew inverse Laurent series rings." Journal of Algebra and Its Applications 16, no. 12 (November 20, 2017): 1750221. http://dx.doi.org/10.1142/s0219498817502218.

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In the present note, we continue the study of skew inverse Laurent series ring [Formula: see text] and skew inverse power series ring [Formula: see text], where [Formula: see text] is a ring equipped with an automorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. Necessary and sufficient conditions are obtained for [Formula: see text] to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, projective-free and [Formula: see text]-ring, respectively. It is shown here that [Formula: see text] (respectively [Formula: see text]) is a domain satisfying the ascending chain condition (Acc) on principal left (respectively right) ideals if and only if so does [Formula: see text]. Also, we investigate the problem when a skew inverse Laurent series ring [Formula: see text] has the same Goldie rank as the ring [Formula: see text] and is proved that, if [Formula: see text] is a semiprime right Goldie ring, then [Formula: see text] is semiprimitive. Furthermore, we study on the relationship between the simplicity, semiprimeness, quasi-Baerness and Baerness property of a ring [Formula: see text] and these of the skew inverse Laurent series ring. Finally, we consider the problem of determining when [Formula: see text] is nilpotent.

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45

Sardanashvily, G., and W. Wachowski. "Differential Calculus onN-Graded Manifolds." Journal of Mathematics 2017 (2017): 1–19. http://dx.doi.org/10.1155/2017/8271562.

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The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, overN-graded commutative rings and onN-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and onZ2-graded manifolds. We follow the notion of anN-graded manifold as a local-ringed space whose body is a smooth manifoldZ. A key point is that the graded derivation module of the structure ring of graded functions on anN-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its bodyZ. Accordingly, the Chevalley–Eilenberg differential calculus on anN-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus onN-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds ofN-graded bundles.

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46

Kumar, Deepak, and Gurninder S. Sandhu. "On Commutativity of Semiprime Rings with Multiplicative (Generalized)-derivations." Journal of Mathematics Research 9, no. 2 (March 5, 2017): 9. http://dx.doi.org/10.5539/jmr.v9n2p9.

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47

Mamouni, A., B. Nejjar, and L. Oukhtite. "Differential identities on prime rings with involution." Journal of Algebra and Its Applications 17, no. 09 (August 23, 2018): 1850163. http://dx.doi.org/10.1142/s0219498818501633.

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In this paper, we investigate commutativity of prime rings [Formula: see text] with involution ∗ of the second kind in which generalized derivations satisfy certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Furthermore, we provide an example to show that the restriction imposed on the involution is not superfluous.

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48

Nabiel, H. "Ring subsets that be center-like subsets." Journal of Algebra and Its Applications 17, no. 03 (February 5, 2018): 1850048. http://dx.doi.org/10.1142/s0219498818500482.

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The aim of this paper is to show that certain subsets, which are defined by commutativity conditions involving derivations, generalized derivations and epimorphisms, coincide with the center in prime or semiprime rings.

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49

Et al., Faraj. "On Skew Left n-Derivations with Lie Ideal Structure." Baghdad Science Journal 16, no. 2 (June 2, 2019): 0389. http://dx.doi.org/10.21123/bsj.16.2.0389.

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In this paper the centralizing and commuting concerning skew left -derivations and skew left -derivations associated with antiautomorphism on prime and semiprime rings were studied and the commutativity of Lie ideal under certain conditions were proved.

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50

Et al., Faraj. "On Skew Left n-Derivations with Lie Ideal Structure." Baghdad Science Journal 16, no. 2 (June 2, 2019): 0389. http://dx.doi.org/10.21123/bsj.2019.16.2.0389.

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In this paper the centralizing and commuting concerning skew left -derivations and skew left -derivations associated with antiautomorphism on prime and semiprime rings were studied and the commutativity of Lie ideal under certain conditions were proved.

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