Advances in Ring Theory and Applications

Frontmatter

A Note on Multiplicative (Generalized)-Derivations and Left Sided Ideals in Semiprime Rings

Abstract

Let R be a 2-torsion free semiprime ring with center Z(R) and \(\lambda \) a nonzero left sided ideal of R. Let \(F,G:R\rightarrow R\) be multiplicative (generalized)-derivations associated with the map \(d:R\rightarrow R\) (not necessarily additive nor derivation) and \(H,T:R\rightarrow R\) be any two maps. The main goal of this article is to study identities:

(1)

 \((d(x)F(y)\pm G(y)d(x))\pm (H(x)y+yT(x))=0\) for all \(x,y\in \lambda \);

 

(2)

 \((d(x)F(y)\pm G(y)d(x))\pm (xy\pm yx)\in Z(R)\) for all \(x,y\in \lambda \).

 

Gurninder S. Sandhu, Basudeb Dhara, Sourav Ghosh

On Weakly Generalized Reversible Rings

Abstract

We introduce a notion of weakly generalized reversible rings as a proper generalization of generalized reversible rings. In support, we give examples. We show that every central reversible ring is an example of weakly generalized reversible ring. Also, we study some properties and characterizarions of weakly generalized reversible ring.

Nirbhay Kumar, Avanish Kumar Chaturvedi

Additivity of Multiplicative b-Generalized (Skew) Derivations

Abstract

Let R be a prime ring and Q be the right Martindale quotient ring of R. Then we prove that a map G on R satisfies \(G(xy)=G(x)y+ b\alpha (x)d(y)\) for all \(x,y\in R\) where \(\alpha \) is an automorphism on R and d is an additive map over R, \(b \in Q\), is additive. Moreover, if a map g on R satisfying \(g(xy)=g(x)y + bxd(y)\) for all \(x,y\in R\) and d, b are mentioned above, then g is additive.

Sk Aziz, Om Prakash

Specht Modules and Representations of Symmetric Group

Abstract

The symmetric group plays a key role in many areas of research in addition to mathematics and hence is of great importance. It is well known that representations of a symmetric group can be approached from three different directions: by using results from the general theory of group representations, by employing combinatorial techniques, or via symmetric functions. In this paper, we study the irreducible representations of symmetric groups by using the Specht Modules, which are irreducible sub-modules of the permutation modules. We provide a complete description of the construction of the irreducible representations via Young Symmetrizers (also called row and column stabilizers) and the Frobenius Formula. We explore a connection between Young Tableaux and the representation of the symmetric group \(S_n\). We describe the construction of Specht Modules \(S^\lambda \) for each partition \(\lambda \) of n.

Maha Oudah Alshammari, Faryad Ali

Commutativity Theorems on Prime Rings with Generalized Derivations

Abstract

Suppose that \(\mathcal {R}\) is a prime ring, \(\mathcal {I}\) a nonzero ideal of \(\mathcal {R}\), \(\mathcal {F}\) a generalized derivation of \(\mathcal {R}\) and n a fixed positive integer. If

$$ (\mathcal {F}(x_1)x_2+x_1\mathcal {F}(x_2)+\mathcal {F}(x_2)x_1+x_2\mathcal {F}(x_1))^n-(x_1x_2+x_2x_1)=0, $$

for all \(x_1,x_2\in \mathcal {I}\), then one of the following holds:

1.

\(\mathcal {R}\) is commutative;

 

2.

\(n=1\) and there exists \(\lambda \in \mathcal {C}\) such that \(\mathcal {F}(x)=\lambda x\), for all \(x\in \mathcal {R}\) with \(2\lambda =1\).

 

If \(char(\mathcal {R})\ne 2\) and

$$ (\mathcal {F}(x_1)x_2+x_1\mathcal {F}(x_2)+\mathcal {F}(x_2)x_1+x_2\mathcal {F}(x_1))^n-(x_1x_2+x_2x_1)\in \mathcal {Z}(\mathcal {R}), $$

for all \(x_1,x_2\in \mathcal {I}\), then one of the following holds:

1.

\(\mathcal {R}\) is commutative.

 

2.

\(n=1\) and there exists \(\lambda \in \mathcal {C}\) such that \(\mathcal {F}(x)=\lambda x\), for all \(x\in \mathcal {R}\) with \(2\lambda =1\).

 

We examine the aforementioned identities in semiprime rings and also obtain some range inclusion results on Banach algebras.

Basudeb Dhara, Sukhendu Kar, Kalyan Singh

Commutativity of -Prime Rings with Generalized Derivation

Abstract

The main purpose of this paper is to work on commutativity of \( \sigma \)-prime rings with second kind involution \( \sigma \), involving generalized derivation satisfy the certain differential identities.

Adnan Abbasi, Muzibur Rahman Mozumder

A Note on -Generalized Skew Derivations on Prime Rings

Abstract

Let \(\mathcal {R}\) be a prime ring with a characteristic not equal to 2. Let \(\mathcal {U}\) and \(\mathcal {C}\) denote its Utumi quotient ring and extended centroid, respectively. Consider a non-central multilinear polynomial \(\phi (\zeta _1, \ldots , \zeta _n)\) over \(\mathcal {C}\), and let \(\textbf{G}\) be a \(\textrm{b}\)-generalized skew derivation of \(\mathcal {R}\), satisfying the identity:

$$ p \phi (\zeta )\textbf{G}(\phi (\zeta ))= \textbf{G}(\phi (\zeta )^2),\ p\ne 2,\ \forall \ \zeta =(\zeta _1\ldots ,\zeta _n)\in \mathcal {R}^n.$$

The purpose of this paper is to classify all potential forms of the \(\textrm{b}\)-generalized skew derivation \(\textbf{G}.\)

Mani Shankar Pandey, Ashutosh Pandey

Analysis of Some Topological Indices Over the Weakly Zero-Divisor Graph of the Ring

Abstract

In this paper, we study some basic properties of weakly zero-divisor graph of the ring \(\mathbb {Z}_p \times \mathbb {Z}_q \times \mathbb {Z}_r\), denoted by \(\text {W}{\Gamma }(\mathbb {Z}_p \times \mathbb {Z}_q \times \mathbb {Z}_r)\), where p, q and r are prime numbers greater than 2 and not necessarily distinct. Further, we discuss Wiener index, Gutman index, first K Banhatti index, second K Banhatti index, Forgotten topological index, Forgotten topological coindex, and degree distance index of \(\text {W}{\Gamma }(\mathbb {Z}_p \times \mathbb {Z}_q \times \mathbb {Z}_r)\).

Nadeem ur Rehman, Shabir Ahmad Mir, Mohd Nazim

A Study of Central Identities Equipped with Skew Lie Product Involving Generalized Derivations

Abstract

Let R be a ring with involution \(\sigma \). Notation \( \nabla [t_1,t_2]\) denotes the skew Lie product and defined by \(t_1 t_2-t_2 \sigma (t_1)\). The main objective of this paper is to investigate the commutativity of \(\sigma \)-prime rings with involution \(\sigma \) of the second kind equipped with skew Lie product involving generalized derivation. Finally, we provide some examples to demonstrate that the conditions assumed in our results are necessary.

Md. Arshad Madni, Mohd Shuaib Akhtar, Muzibur Rahman Mozumder

Some Results on Left-sided Ideals of Semiprime Rings with Symmetric n-Derivations

Abstract

Let R be a prime or semiprime ring and \(\alpha ,\beta \) are automorphisms of R. An n-additive mapping \(\Delta : R^n \rightarrow R\) is said to be \((\alpha ,\beta )\) n-derivation if \(\Delta (h_1,h_2,...,\) \(x_ix_i',...,h_n) = \Delta (h_1,h_2,...,x_i,...,h_n)\alpha (x_i') + \beta (x_i)\Delta (h_1,h_2,...,x_i',...,h_n)\). In the present paper, we shall prove that the map \(\Delta :R \times R \times ...\times R \rightarrow R\) is zero if it satisfies the identity \([\delta (x), \beta (x)] = 0\) and \(\delta (x) \circ \beta (x)= 0\) for all \(x \in I\), where I is a left ideal of R and \(\delta \) be the trace of \(\Delta \). This result is also the generalization of Fošner result [7, Theorem 1].

Muzibur Rahman Mozumder, Nazia Parveen, Wasim Ahmed

Central Power Values of Generalized Derivation and Structure of Unital Banach Algebra

Abstract

The structure of a prime Banach algebra \(\mathfrak {A}\) containing the identity element e with the help of a nonzero continuous linear generalized derivation \(\mathfrak {D}\) satisfying one of the following conditions, is investigated in this paper : (i) \(x\Big (\mathfrak {D}((ab)^m)\pm a^mb^m\Big )\in \mathcal {Z}_\mathfrak {A}\); (ii) \(x\Big (\mathfrak {D}((ab)^m)\pm b^ma^m\Big )\in \mathcal {Z}_\mathfrak {A}\); (iii) \(x\Big (\mathfrak {D}((ab)^m)\pm \mathfrak {D}(a^mb^m)\Big )\in \mathcal {Z}_\mathfrak {A}\); (iv) \(x\Big (\mathfrak {D}((ab)^m)\pm \mathfrak {D}(b^ma^m)\Big )\in \mathcal {Z}_\mathfrak {A}\); (v) \(x\Big (\mathfrak {D}((ab)^m)\pm [a^m, b^m]\Big )\in \mathcal {Z}_\mathfrak {A}\); (vi) \(x\Big (\mathfrak {D}((ab)^m)\pm [b^m, a^m]\Big )\in \mathcal {Z}_\mathfrak {A}\); (vii) \(x\Big (\mathfrak {D}((ab)^m)\pm (ab)^m\Big )\in \mathcal {Z}_\mathfrak {A}\); (viii) \(x\Big (\mathfrak {D}((ab)^m)\Big )\in \mathcal {Z}_\mathfrak {A}\) for all a, b in some suitable non empty subset of \(\mathfrak {A}\), where m is a positive integer such that \(m=m(a, b)>1\). Moreover, we give a characterization of the map \(\mathfrak {D}\). Finally it is shown that our theorems do not hold in case \(\mathfrak {A}\) is not prime.

Asma Ali, Kapil Kumar, Mohd Tasleem

Generalized Skew Derivation of Order 2 in Prime Ring with Multilinear Polynomial

Abstract

Let \(\mathcal {R}\) be a prime ring of characteristic not equal to 2, \(\mathcal {U}\) be the Utumi quotient ring of \(\mathcal {R}\), \(\mathcal {C}\) be the extended centroid of \(\mathcal {R}\) and \(f(\zeta _1, \ldots ,\zeta _n)\) be a non-central polynomial identity of \(\mathcal {R}\). Let \(\mathcal {H}\), \(\mathcal {F}\) are generalized skew derivations on \(\mathcal {R}\) such that \(\mathcal {F}(\mathcal {F}(u)u) = \mathcal {H}(u^2)\) for all \(u =f(\zeta _1,\ldots ,\zeta _n)\), \(\zeta _1,\zeta _2,\dots ,\zeta _n\in \mathcal {R}\). In this article, we study the complete structure of \(\mathcal {F}\) and \(\mathcal {H}\).

Ashutosh Pandey, Balchand Prajapati

Local and 2-local Lie-type Derivations of Operator Algebras on Banach Spaces

Abstract

Let X be a Banach space over the field \(\mathbb {F}\) (\(\mathbb {F}\) is either the real field \(\mathbb {R}\) or the complex field \(\mathbb {C}\)). Let B(X) be the set of all bounded linear operators on X and F(X) be the set of all finite rank operators in B(X). A subalgebra \(\mathcal {A}\) of B(X) is called a standard operator algebra if \(\mathcal {A}\) contains F(X). Suppose that \(\delta \) is a map from \(\mathcal {A}\) into B(X). Firstly, we prove that if \(\delta \) is a Lie-type derivation, then \(\delta \) has the standard form. Furthermore, we show that if \(\delta \) is a local Lie-type derivation, then \(\delta \) is a Lie-type derivation. Finally, we prove that if \(\delta \) is a 2-local Lie n-derivation, then \(\delta =d+\tau \), where d is a derivation, and \(\tau \) is homogeneous map from \(\mathcal {A}\) into \(\mathbb {F}I\) such that \(\tau (A+B)=\tau (A)\) for each A, B in \(\mathcal {A}\) where B is a sum of \((n-1)\)-th commutators.

Zhi-Cheng Deng, Feng Wei

The Noncommutative Singer-Wermer Conjecture and Generalized Skew Derivations

Abstract

The noncommutative Singer-Wermer conjecture states that every linear derivation on a noncommutative Banach algebra maps into its Jacobson radical. This conjecture is still an open question for more than thirty years. In this paper, the question of when a generalized skew derivation on a Banach algebra has quasinilpotent values is considered and how this question is related to the noncommutative Singer-Wermer conjecture is discussed.

Feng Wei, Jing-Xiong Xu

Non-global Multiplicative Lie Triple Derivations on Rings

Abstract

Let \(\mathfrak {R}\) be a ring containing a nontrivial idempotent with center \(\mathcal {Z}(\mathfrak {R})\). In the present article, it is shown that under certain restrictions every map \(\xi :\mathfrak {R}\rightarrow \mathfrak {R}\) (not necessarily additive) satisfying \(\xi ([[S, T], U])=[[\xi (S), T], U]+[[S,\xi (T)],\) \( U]+[[S, T],\xi (U)]\) for all \(S, T, U\in \mathfrak {R}\) with \(STU=0,\) is almost additive, that is, \(\xi (S+T)-\xi (S)-\xi (T)\in \mathcal {Z}(\mathfrak {R}).\) In addition, if \(\mathfrak {R}\) is a 2-torsion free prime ring, then \(\xi \) is of the form \(\xi =\partial +\eta ,\) where \(\partial \) is a derivation from \(\mathfrak {R}\) into its central closure \(\mathfrak {S}\) and \(\eta \) is a map from \(\mathfrak {R}\) into its extended centroid \(\mathfrak {C}\) such that \(\eta (S+T)-\eta (S)-\eta (T)\in \mathcal {Z}(\mathfrak {R})\) and \(\eta ([[S, T], U])=0\) for all \(S,T,U\in \mathfrak {R}\) with \(STU=0.\) The obtained results are then applied to standard operator algebras, factor von Neumann algebras and the algebra of all bounded linear operators.

Mohammad Ashraf, Mohammad Afajal Ansari, Md Shamim Akhter

Algebraic Identities on Prime and Semiprime Rings

Abstract

The goal of this scientific research is to show that an additive mapping \(\mathcal {T}\) from a prime ring R to itself is a centralizer with an appropriate torsion restriction on R if it fulfills some algebraic identities. Furthermore, certain real instances are used to validate these findings.

Abu Zaid Ansari, Faiza Shujat

Generalized Derivations with Periodic Values on Prime Rings

Abstract

Let R be a prime ring whose characteristic is either zero or \(p>0\) such that \(p(2^n-2)\). Let \(Q_r\) be its right Martindale quotient ring, C its extended centroid and F and G non-zero generalized derivations of R. If

$$ \biggl (F(x)x-xG(x)\biggr )^n=\biggl (F(x)x-xG(x)\biggr ) $$

for all \(x\in [R,R]\), with \(n\ge 2\) fixed integer, then one of the following holds:

1.

there exists \(a\in Q_r\) such that \(F(x)=xa\) and \(G(x)=ax\), for all \(x\in R\);

 

2.

\(R\subseteq M_2(C)\), the ring of all \(2\times 2\) matrices over C, and there exist \(a,c\in R\) such that \(F(x)=ax+xc\) and \(G(x)=cx+xa\), for all \(x\in R\);

 

3.

\(R\subseteq M_2(C)\), the ring of all \(2\times 2\) matrices over C, and there exist \(a,b,q\in R\) such that \(F(x)=ax+xb\) and \(G(x)=bx+xq\), for all \(x\in R\), where \((a-q)^n=a-q\). Moreover C is periodic and, for all \(x \in [R,R]\), \(x^{2n}=x^2\).

 

Giovanni Scudo

On a Functional Identity Involving Power Values of Generalized Skew Derivations on Lie Ideals

Abstract

Let R be a prime ring, \(Q_r\) its right Martindale quotient ring and C its extended centroid. Suppose that F is a generalized skew derivation of R, L a non-central Lie ideal of R, \(m,n,s\ge 1\) positive fixed integers. If

$$ F(u)^n\biggl (F(u)^m-u^m\biggr )^s=0 $$

for all \(u\in L\), then there exists \(\lambda \in C\) such that \(F(x)=\lambda x\), for any \(x\in R\), with \(\lambda ^m=1\), unless when \(char(R)=2\) and \(R\subseteq M_2(K)\), the ring of \(2\times 2\) matrices over a field K. We will also provide a generalization of the previous result for semiprime rings.

Luisa Carini, Vincenzo De Filippis

Generalized Skew Derivations with Periodic Values on Lie Ideals

Abstract

Let R be a prime non commutative ring, L a noncentral Lie ideal of R, \(\alpha \) an automorphism of R, \(G: R\longrightarrow R\) a non-zero generalized skew derivation of R, \(m,n\ge 2\) fixed positive integers, such that

$$\begin{aligned}\biggl (G(u)^n-G(u)\biggr )^m=0,\,\,\,\text { for all }u\in L.\end{aligned}$$

If \(char(R)=0\) or \(char(R)=p>0\) where \(p \not \mid 2^{n}-2\), then \(R\subseteq M_2(C)\), the ring of \(2\times 2\) matrices over the extended centroid C.

Milena Andaloro

Generalized Skew-Derivations Acting on Multilinear Polynomials in Prime Rings

Abstract

Let R be a noncommutative prime ring of characteristic different from 2, \(Q_r\) be its right Martindale quotient ring, C be its extended centroid and \(f(r_1,\ldots ,r_n)\) be a noncentral multilinear polynomial over C. Suppose that \(T_1\), \(T_2\) are two generalized skew derivations on R. If

$$\begin{aligned} T_1(u)T_2(u)=T_1(u)u-uT_2(u)\end{aligned}$$

for all \(u\in f(R)=\{f(x_1,\ldots ,x_n) | x_1,\ldots ,x_n\in R\}\), then we determine all possible forms of the maps.

Manami Bera, Basudeb Dhara

Results on b-Generalized Derivations in Rings

Abstract

This paper studies various identities of noncommutative prime rings involving b-generalized derivations. In particular, we prove that if a noncommutative prime ring \(\mathfrak {R}\) admits a nonzero b-generalized derivation \(\mathfrak {F}\) such that \([\mathfrak {F}(l), m]=[l, \mathfrak {F}(m)]\) for all \(l,m\in \mathfrak {I}\), where \(\mathfrak {I}\) is a nonzero ideal of \(\mathfrak {R}\), then \(\mathfrak {F}(l)=a l\) for all \(l\in \mathfrak {R}\) and for some \(a\in \mathcal {C}\) unless \(\mathfrak {R}\subseteq M_2(F)\), the \(2\times 2\) matrix ring over the field F. This gives a natural generalization of the results for derivations, generalized derivations and generalized \(\rho \)-derivations with an X-inner automorphism.

Mohammad Salahuddin Khan, Shakir Ali, Abdul Nadim Khan, Mohammed Ayedh

Results on Generalized Skew Bi-Semiderivation in Prime Rings

Abstract

Let \(\mathcal {R}\) be a prime ring of \(char\ne 2, 3\), \(\mathcal {D}\) a symmetric skew bi-semiderivation of \(\mathcal {R}\) associated with automorphism \(\alpha \) and \(\varDelta \) be a symmetric generalized skew bi-semiderivation associated with \(\alpha , f, \mathcal {D}\), where f is surjective map and \(\alpha \) is an automorphism. If \(x\varDelta (x,x)=0\) for all \(x\in \mathcal {R}\), then \(\varDelta =0\).

Faiza Shujat, Abu Zaid Ansari

Commuting Iterates of Generalized Derivations on Lie Ideals

Abstract

Let R be a prime ring of characteristic different from 2, \(Q_r\) its right Martindale quotient ring, C its extended centroid, F and G two nonzero generalized derivations of R, L a non-central Lie ideal of R and \(n \ge 1\) a fixed integer. If

$$\begin{aligned} \Bigl \{F^2(x)x-xG^2(x)\Bigr \}^n=0 \end{aligned}$$

for all \(x \in L\), then either \(R \subseteq M_2(K)\), the ring \(2 \times 2\) matrices over a field K, or one ot the following holds:

1.

\(F(x) = xa\) and \(G(x) = xc\) for all \(x \in R\) with \(a^2 = c^2 \in C\);

 

2.

\(F(x) = xa\) and \(G(x) = cx\) for all \(x \in R\) with \(a^2 = c^2\);

 

3.

\(F(x) = ax\) and \(G(x) = xc\) for all \(x \in R\) with \(a^2 = c^2 \in C\);

 

4.

\(F(x) = ax\) and \(G(x) = cx\) for all \(x \in R\) with \(a^2 = c^2 \in C\).

 

Francesco Ammendolia

Non-linear Mappings Preserving Product on Factor von Neumann Algebras

Abstract

In this manuscript, we explore the form of a non-linear bijective maps on factor von Neumann algebras \(\textsf{M}\) and \(\textsf{N}\) such that \(\varOmega :\textsf{M}\rightarrow \textsf{N}\) satisfies \(\varOmega (\texttt {m} ~\diamondsuit ~ \texttt {n})=\varOmega (\texttt {n})~\diamondsuit ~\varOmega (\texttt {m})\), where \(\texttt {m}~\diamondsuit ~ \texttt {n}=\texttt {m}^{*}{} \texttt {n}+\texttt {n}^{*}{} \texttt {m}\), for all \(\texttt {m},\texttt {n}\in \textsf{M}\).

Mohd Arif Raza, Tahani Al-Sobhi

A Commutativity Condition for Semiprime Rings with Generalized Skew Derivations

Abstract

Let R be a semiprime ring of characteristic different from 2, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, F a generalized skew derivation of R and \(n\ge 1\) a fixed integer such that \(\bigr (F(x)y+F(y)x-[x,y]\bigl )^n=0\), for all \(x,y \in R\). Then R is commutative and \(F=0\).

Francesco Rania

Results on Generalized Derivations in Prime Rings with Involution

Abstract

Let \(\mathscr {A}\) be a prime ring with the center \(\mathscr {Z}(\mathscr {A})\) and a second-kind involution \(*\) in which generalized derivations fulfil specific algebraic identities. Assume that \(\psi \) and \(\phi \) are generalized derivations associated with \(\xi \) and \(\mu \) on \(\mathscr {A},\) respectively. In this article, we examine the following situation: (i) \(\psi (x)x^*-x\phi (x^*)\in \mathscr {Z}(\mathscr {A}),\) (ii) \(\psi (x)x^*-x^*\phi (x)\in \mathscr {Z}(\mathscr {A}),\) (iii) \(\psi (x)x-x\phi (x^*)\in \mathscr {Z}(\mathscr {A}),\) (iv) \(\psi (x)x-\) \(x^*\phi (x)\in \mathscr {Z}(\mathscr {A}),\) (v) \(\psi (x)x-x^*\phi (x^*)\in \mathscr {Z}(\mathscr {A}),\) \(\forall \) \(x\in \mathscr {A}.\)

Faez. A. Alqarni, Nadeem ur Rehman, Hafedh M. Alnoghashi

Quantum Codes Over an Extension of

Abstract

Let \(\mathfrak {A}=\mathbb Z_4+u\mathbb Z_4+v\mathbb Z_4,\) where \(u^2=u\), \(v^2=v\) and \(uv=vu=0\) be a ring, which is an extension of \(\mathbb {Z}_{4}\). In this article, we study the structure of cyclic codes over the ring \(\mathfrak {A}\) and define a \(\mathbb Z_2\)-linear isometry \(\Phi \) from \(\mathfrak {A}^{n}\) to \(\mathbb Z^{6n}_2.\) Based on the classical cyclic codes, we construct binary quantum codes by utilizing Gray images of cyclic codes over \(\mathfrak {A}\). As an application, we provide some examples of binary quantum error-correcting codes.

Mohammad Ashraf, Naim Khan, Washiqur Rehman, Ghulam Mohammad

Product of Traces of Permuting n-Derivations on Prime and Semiprime Ideals of a Ring

Abstract

In this paper, we prove that if the product of the traces of two permuting n-derivations on a ring maps the ring into a prime ideal, then one of the traces maps the ring into the prime ideal. We will also extend the above result on the product of two or more traces under suitable characteristic restrictions. In fact, we proved that for a semiprime ideal \(\mathcal {P}\) of \(\mathcal {R}\), \(d^r(\mathcal {P})\subseteq \mathcal {P}\) if and only if \(d(\mathcal {P})\subseteq \mathcal {P}\) for any positive integer r.

Nazia Parveen