Papers by Mohammad Ashraf
In this article, we will cover a concept called multiplicative (generalized) skew derivation on r... more In this article, we will cover a concept called multiplicative (generalized) skew derivation on rings, and we will generalize some of the important results in the literature. After, we enrich this paper with examples which show that our used assumptions are essential.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Archivum Mathematicum, 1997
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
International Journal of Algebra, 2015
Let R be a 2-torsion free prime ring with center Z, right Utumi quotient ring U , generalized der... more Let R be a 2-torsion free prime ring with center Z, right Utumi quotient ring U , generalized derivation F associated with a nonzero derivation d of R and L a Lie ideal of R. If F (uv) n = F (u) m F (v) l or F (uv) n = F (v) l F (u) m for all u, v ∈ L, where m, n, l are fixed positive integers, then L ⊆ Z. We also examine the case where R is Ahmad N. Alkenani et al. a semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on non-commutative Banach algebras.
Journal of Taibah University for Science, 2014
In the present paper it is shown that zero symmetric prime right near-rings satisfying certain id... more In the present paper it is shown that zero symmetric prime right near-rings satisfying certain identities are commutative rings.
Miskolc Mathematical Notes, 2018
Let n 1 be a fixed positive integer and R be a ring. A permuting n-additive map W R n ! R is know... more Let n 1 be a fixed positive integer and R be a ring. A permuting n-additive map W R n ! R is known to be permuting generalized n-derivation if there exists a permuting nderivation W R n ! R such that˝.x 1 ; x 2 ; ; x i x 0 i ; ; x n / D˝.x 1 ; x 2 ; ; x i ; ; x n /x 0 i C x i .x 1 ; x 2 ; ; x 0 i ; ; x n / holds for all x i ; x 0 i 2 R. A mapping ı W R ! R defined by ı.x/ D .x; x; ; x/ for all x 2 R is said to be the trace of. The trace ! of˝can be defined in the similar way. The main result of the present paper states that if R is a .n C 1/Š-torsion free semi-prime ring which admits a permuting n-derivation such that the trace ı of satisfies OEOEı.x/; x; x 2 Z.R/ for all x 2 R; then ı is commuting on R. Besides other related results it is also shown that in a nŠ-torsion free prime ring if the trace ! of a permuting generalized n-derivation˝is centralizing on R; then ! is commuting on R.
Gulf Journal of Mathematics
Let R be an associative ring with involution ∗. In this paper we introduce the notion of (α,β)∗-n... more Let R be an associative ring with involution ∗. In this paper we introduce the notion of (α,β)∗-n-derivation in R, where α and β are endomorphisms of R. An additive mapping x↦x∗ of R into itself is called an involution on R if it satisfies the conditions: (i) (x∗)∗=x, (ii) (xy)∗=y∗x∗ for all x,y∈ R. A ring R equipped with an involution ∗ is called a ∗-ring. In the present paper it is shown that if a ∗-prime ring R admits a nonzero (α,β)∗-n-derivation D such that α is surjective, then R is commutative. Some properties of certain n-additive mappings are also discussed in the setting of ∗-prime rings and semiprime ∗-rings. Further, some related properties of (α,β)∗-n-derivation in semiprime ∗-ring have also been investigated. Besides, we have also constructed several examples throughout the text to justify that various restrictions imposed in the hypotheses of our theorems are not superfluous. Finally a structure theorem for (α,β)∗-n-derivation in a semiprime ∗-ring has been established.
Annals of the Alexandru Ioan Cuza University - Mathematics
Let U be a triangular algebra. Under certain assumptions, we show that every multiplicative Lie σ... more Let U be a triangular algebra. Under certain assumptions, we show that every multiplicative Lie σ-derivation L : U → U is of the form δ + τ, where δ is an additive σ-derivation on U and τ is a map from U to its center Zσ(U) that vanishes on Lie product. As an application, we study the multiplicative Lie σ-derivation on nest algebras and block upper triangular matrix algebras.
Boletim da Sociedade Paranaense de Matemática, 2022
Let R be a semiprime (or prime) ring and U be a nonzero ideal of R. In the present paper, we stud... more Let R be a semiprime (or prime) ring and U be a nonzero ideal of R. In the present paper, we study the notions of multiplicative generalized α-skew derivations on ideals of R and prove that if R admits a multiplicative generalized α-skew derivation G associated with a nonzero additive map d and an automorphism α, then d is necessarily an α-skew derivation of R. Also, we study the structure of a semiprime ring admitting a multiplicative generalized α-skew derivation satisfying more specific algebraic identities. Moreover, we also provide examples to show that the assumed restrictions cannot be relaxed
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2020
Let be a commutative ring with unity, 𝒜, be -algebras, be (𝒜, )-bimodule and 𝒩 be (, 𝒜)-... more Let be a commutative ring with unity, 𝒜, be -algebras, be (𝒜, )-bimodule and 𝒩 be (, 𝒜)-bimodule. The -algebra 𝒢 = 𝒢(𝒜, , 𝒩, ) is a generalized matrix algebra defined by the Morita context (𝒜, , , 𝒩, ξ𝒩, Ω𝒩). In this article, we study Jordan σ-derivations on generalized matrix algebras.
Mathematics, 2021
In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring R ... more In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring R with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three disjoint subsets of R. After that, connectivity, clique number, and girth have also been studied. Finally, we determine the cases when it is Eulerian, Hamiltonian, and contains a Eulerian trail.
Contributions Discret. Math., 2020
Let $\mathscr{W}$ be a fixed $k$-dimensional subspace of an $n$-dimensi\-onal vector space $\math... more Let $\mathscr{W}$ be a fixed $k$-dimensional subspace of an $n$-dimensi\-onal vector space $\mathscr{V}$ such that $n-k\geq1.$ In this paper, we introduce a graph structure, called the subspace based subspace inclusion graph $\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V}),$ where the vertex set $\mathscr{V}(\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V}))$ is the collection of all subspaces $\mathscr{U}$ of $\mathscr{V}$ such that $\mathscr{U}+\mathscr{W}\neq\mathscr{V}$ and $\mathscr{U}\nsubseteq\mathscr{W},$ i.e., $\mathscr{V}(\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V}))= \{\mathscr{U}\subseteq\mathscr{V}~|~\mathscr{U}+\mathscr{W}\neq\mathscr{V}, \mathscr{U}\nsubseteq\mathscr{W}\}$ and any two distinct vertices $\mathscr{U}_{1}$ and $\mathscr{U}_{1}$ of $\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V})$ are adjacent if and only if either $\mathscr{U}_{1}+\mathscr{W}\subset\mathscr{U}_{2}+\mathscr{W}$ or $\mathscr{U}_{2}+\mathscr{W}\subset\mathscr{U}_{1}+\mathscr{W}.$ The diameter, girth, clique n...
Let M be a Γ-ring and N be the set of non-negative integers. A family D = {d n } n∈N of additive ... more Let M be a Γ-ring and N be the set of non-negative integers. A family D = {d n } n∈N of additive mappings d n : M → M such that d 0 = I M is said to be a triple higher derivation (resp. Jordan triple higher derivation) on M if d n (aαbβc) = p+q+r=n d p (a)αd q (b)βd r (c) (resp. d n (aαbβa) = p+q+r=n d p (a)αd q (b)βd r (a)) holds for all a, b, c ∈ M and α, β ∈ Γ, and for each n ∈ N. In the present paper it is shown that on prime Γ-ring M of characteristic different from two every Jordan triple higher derivation on M is a higher derivation on M .
Motivated by the works of Wang [Y. Wang, Lie (Jordan) derivations of arbitrary triangular algebra... more Motivated by the works of Wang [Y. Wang, Lie (Jordan) derivations of arbitrary triangular algebras, Aequationes Mathematicae, 93 (2019), 1221-1229] and Moafian et al. [F. Moafian and H. R. Ebrahimi Vishki, Lie higher derivations on triangular algebras revisited, Filomat, 30(12) (2016), 3187-3194.], we shall study Lie higher derivations of arbitrary triangular algebras. In fact, it is shown that every Lie higher derivation on an arbitrary triangular algebra is proper, using the notion of maximal left (right) ring of quotients. 2010 Mathematics Subject Classification: 16W25, 15A78, 16R60
In the present paper, it is shown that a semi-prime ring R of characteristic not 2 and 3 contains... more In the present paper, it is shown that a semi-prime ring R of characteristic not 2 and 3 contains a non-zero central ideal of R, if R admits an automorphism ζ such that [s , w] = [s , w] for every s, w ∈ R, where 1 < m ∈ Z. We shall also study the case when the underlying condition holds for the elements from a non-cental Lie ideal of a prime ring R. The latter result is in the spirit of Herstein’s theorem which deals with the commutator having idempotent values in rings.
Let R be a prime ring with center Z(R). Suppose that R admits a generalized semiderivation F with... more 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.
In this paper, we study the extended zero divisor graph EG(S) associated to a commutative semigro... more In this paper, we study the extended zero divisor graph EG(S) associated to a commutative semigroup S with zero. Also, we characterize all extended zero divisor graphs with three vertices.
We present an historical account of the study of derivations, generalized derivations, n-derivati... more We present an historical account of the study of derivations, generalized derivations, n-derivations, generalized n-derivation and other kinds of derivations in near-rings, based on the work of several authors. Moreover, recent results on semigroup ideals and generalized n-derivations on these topics have been discussed in details. Examples of various notions have also been included.
Advances in Pure and Applied Mathematics, 2020
Let R be a *-ring with the center Z(R) and N be the set of all non-negative integers. Let L = {L ... more Let R be a *-ring with the center Z(R) and N be the set of all non-negative integers. Let L = {L n } n∈N be the family of mappings L n : R → R (not necessarily additive) such that L 0 = I R , the identity mapping of R. Then L is said to be a *-Lie higher derivable mapping of R if L n ([X * , Y ]) = i+j=n [L i (X) * , L j (Y)] holds for all X, Y ∈ R. In this paper, it is shown that, if R is a *-ring containing a nontrivial self adjoint idempotent which admits a *-Lie higher derivable mapping L = {L n } n∈N , then there exists an element Z X,Y (depending on X and Y) in the center Z(R) such that L n (X + Y) = L n (X) + L n (Y) + Z X,Y .
Rad Hrvatske akademije znanosti i umjetnosti. Matematičke znanosti, 2019
Let R be a *-ring containing a nontrivial self-adjoint idempotent. In this paper it is shown that... more Let R be a *-ring containing a nontrivial self-adjoint idempotent. In this paper it is shown that under some mild conditions on R, if a mapping d : R → R satisfies d([U * , V ]) = [d(U) * , V ] + [U * , d(V)] for all U, V ∈ R, then there exists Z U,V ∈ Z(R) (depending on U and V), where Z(R) is the center of R, such that d(U + V) = d(U) + d(V) + Z U,V. Moreover, if R is a 2-torsion free prime *-ring additionally, then d = ψ + ξ, where ψ is an additive *-derivation of R into its central closure T and ξ is a mapping from R into its extended centroid C such that ξ(U + V) = ξ(U) + ξ(V) + Z U,V and ξ([U, V ]) = 0 for all U, V ∈ R. Finally, the above ring theoretic results have been applied to some special classes of algebras such as nest algebras and von Neumann algebras.
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Papers by Mohammad Ashraf