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Let [Formula: see text], where p is an odd prime and r is a positive integer. Consider a ring [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text] with [Formula: see text]. In this paper, we examine the algebraic structure of constacyclic codes over the ring [Formula: see text] of block length [Formula: see text]. Further, a construction of quantum error-correcting codes (QECCs) from constacyclic codes over [Formula: see text] is also given. Moreover, we derive a number of new QECCs.

Let $ \mathcal{A} $ be a unital $ \ast $-algebra containing a non-trivial projection. In this paper, we prove that if a map $ \Omega $ : $ \mathcal{A} $ $ \to $ $ \mathcal{A} $ such that$ \begin{equation} \nonumber \Omega( [ \mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D}) = [\Omega(\mathscr{K}), \mathscr{F}]_\ast \odot \mathscr{D} + [ \mathscr{K}, \Omega (\mathscr{F})]_\ast \odot \mathscr{D} + [ \mathscr{K}, \mathscr{F}]_\ast \odot \Omega (\mathscr{D}), \end{equation} $where $ [\mathscr{K}, \mathscr{F}]_{\ast} = \mathscr{K}\mathscr{F}- \mathscr{F}\mathscr{K}^\ast $ and $ \mathscr{K} \odot \mathscr{F} = \mathscr{K}^\ast \mathscr{F}+ \mathscr{F}\mathscr{K}^\ast $ for all $ \mathscr{K}, \mathscr{F}, \mathscr{D} \in \mathcal{A}, $ then $ \Omega $ is an additive $ \ast $-derivation. Furthermore, we extend its results on factor von Neumann algebras, standard operator algebras and prime $ \ast $-algebras. Additionally, we provide an example illustrating the existence of such maps.

In this article, we find the Randić spectrum of the weakly zero-divisor graph of a finite commutative ring [Formula: see text] with identity [Formula: see text], denoted as [Formula: see text], where [Formula: see text] is taken as the ring of integers modulo [Formula: see text]. The weakly zero-divisor graph of the ring [Formula: see text] is a simple undirected graph with vertices representing non-zero zero-divisors in [Formula: see text]. Two vertices, denoted as a and b, are connected if there are elements x in the annihilator of a and y in the annihilator of b such that their product xy equals zero. In particular, we examine the Randić spectrum of [Formula: see text] for specific values of [Formula: see text], which are products of prime numbers and their powers.

The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring R=Fp+w1Fp+w2Fp+w22Fp+w1w2Fp+w1w22Fp, where w12−α2=0, w1w2=w2w1, w23−β2w2=0, and α,β∈Fp∖{0}, for a prime p. We begin by introducing a Gray map defined over R, which is associated with an invertible matrix. We demonstrate its advantages over the canonical Gray map through some examples. Finally, we create new and improved quantum codes from constacyclic codes over R using Calderbank–Shore–Steane (CSS) construction.

This paper's primary goal is to look at a quotient ring $\mathscr{A}/\mathscr{T}$ structure, where $\mathscr{A}$ is an arbitrary ring and $\mathscr{T}$ is a semi-prime ideal of $\mathscr{A}$. More precisely, we examine the differential identities in a semi-prime ideal of an arbitrary ring involving $\mathscr{T}$-commuting generalized derivation. Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.

This paper investigates the relationship between the commutativity of rings and the properties of their multiplicative generalized derivations. Let F be a ring with a semiprime ideal Π. A map ϕ:F→F is classified as a multiplicative generalized derivation if there exists a map σ:F→F such that ϕ(xy)=ϕ(x)y+xσ(y) for all x,y∈F. This study focuses on semiprime ideals Π that admit multiplicative generalized derivations ϕ and G that satisfy certain differential identities within F. By examining these conditions, the paper aims to provide new insights into the structural aspects of rings, particularly their commutativity in relation to the behavior of such derivations.

This paper examines the commutativity of the quotient ring F/Y by utilizing specific differential identities in a general ring F that contains a semiprime ideal Y. This study particularly focuses on the role of a multiplicative generalized semiderivation ψ, which is associated with a map θ, in determining the commutative nature of the quotient ring.

We introduce center-like subsets Z∘*(A,d),Z∘**(A,d), where A is the ring and d is the multiplicative derivation. In the following, we take a new derivation for the center-like subsets existing in the literature and establish the relations between these sets. In addition to these new sets, the theorems are generalized as multiplicative derivations instead of the derivations found in previous studies. Additionally, different proofs are provided for different center-like sets. Finally, we enrich this article with examples demonstrating that the hypotheses we use are necessary.

This paper concentrates on examining the characterization of nonlinear mixed bi-skew Lie triple ∗- derivations within an ∗-algebra denoted by A which contains a nontrivial projection with a unit I. Additionally, we expand this investigation to applications by describing these derivations within prime ∗-algebras, von Neumann algebras, and standard operator algebras.

For a finite commutative ring R with identity 1≠0, the weakly zero-divisor graph of R denoted as WΓ(R) is a simple undirected graph having vertex set as a set of non-zero zero-divisors of R and two distinct vertices a and b are adjacent if and only if there exist elements r∈ann(a) and s∈ann(b) satisfying the condition rs=0. The zero-divisor graph of a ring is a spanning sub-graph of the weakly zero-divisor graph. This article finds the normalized Laplacian spectra of the weakly zero-divisor graph WΓ(R). Specifically, the investigation is carried out on the weakly zero-divisor graph WΓ(Zn) for various values of n.

Let R be a commutative ring with identity, and Z(R) be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by WΓ(R) is an undirected (simple) graph with vertex set Z(R)*, and two distinct vertices x and y are adjacent, if and only if there exist r∈ann(x) and s∈ann(y), such that rs=0. Importantly, it is worth noting that WΓ(R) contains the zero-divisor graph Γ(R) as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring Zp×Zt×Zs, denoted by WΓ(Zp×Zt×Zs), where p, t, and s are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of WΓ(Zp×Zt×Zs).

Let A be any arbitrary associative ring, P a semiprime ideal, and J a nonzero ideal of A. In this study, using multiplicative (generalized)-derivations, we explore the behavior of semiprime ideals that satisfy certain algebraic identities. Moreover, examples are provided to demonstrate that the restrictions imposed on the hypotheses of the various theorems are necessary.

Let ℧ be a prime ring of char(℧) ≠2 with its center Z. This article introduces new classes of endomorphisms and investigates how they relate to antiautomorphisms of prime rings and the commutativity of prime rings. Additionally, we fully describe and classify some of these endomorphisms. We also give examples to prove the necessity of the numerous restrictions included in the hypotheses of our results.

This article examines the commutativity of rings with antiautomorphisms, specifically when they are equipped with derivations that satisfy certain algebraic identities. Moreover, we present examples to demonstrate the necessity of the various restrictions imposed in the hypotheses of our theorems.

Let A be a non-commutative prime ring with involution ∗, of characteristic ≠2(and3), with Z as the center of A and Π a mapping Π:A→A such that [Π(x),x]∈Z for all (skew) symmetric elements x∈A. If Π is a non-zero CE-Jordan derivation of A, then A satisfies s4, the standard polynomial of degree 4. If Π is a non-zero CE-Jordan ∗-derivation of A, then A satisfies s4 or Π(y)=λ(y−y*) for all y∈A, and some λ∈C, the extended centroid of A. Furthermore, we give an example to demonstrate the importance of the restrictions put on the assumptions of our results.

In this article, we investigate the formation of reversible cyclic codes (i.e., its codewords forms a symmetry) over the ring S=F2+uF2+u2F2, where u3=0. We find a unique set of generators for cyclic codes over S and classify reversible cyclic codes to their generators. The dual reversible cyclic codes are studied as well. Moreover, we provide some examples of reversible cyclic codes.

Let A be a commutative ring with unity and let set of all zero divisors of A be denoted by ZA. An ideal ℐ of the ring A is said to be essential if it has a nonzero intersection with every nonzero ideal of A. It is denoted by ℐ≤eA. The generalized zero-divisor graph denoted by ΓgA is an undirected graph with vertex set ZA∗ (set of all nonzero zero-divisors of A) and two distinct vertices x1 and x2 are adjacent if and only if annx1+annx2≤eA. In this paper, first we characterized all the finite commutative rings A for which ΓgA is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings A for which ΓgA is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of ΓgA.

Let S be a commutative ring with unity, and a set of nonunit elements is denoted by WS. The coannihilator graph of S, denoted by AG′S, is an undirected graph with vertex set WS∗ (set of all nonzero nonunit elements of S), and α∼β is an edge of AG′S⇔α∉αβS or β∉αβS, where δS denotes the principal ideal generated by δ∈S. In this study, we first classify finite ring S, for which AG′S is isomorphic to some well-known graph. Then, we characterized the finite ring S, for which AG′S is toroidal or projective.

In this article, we study multiplicative (generalized)-skew derivation G and multiplicative left centralizer H satisfying certain conditions in semiprime rings. Moreover, some examples are given to demonstrate that the semiprimeness imposed on the hypotheses of various theorems is essential.

Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators of X and let A(X)⊆L(X) be a standard operator algebra. Suppose there exists a linear mapping T:A(X)→L(X) satisfying the relation T(An) = T(A)An−1 − AT(An−2)A − An−1T(A) for all A∈A(X), where n > 2 is some fixed integer. Then T is of the form: (i)T(A) = 0 for all A∈F(X) and (ii) T(A) = BA, for all A∈A(X) and some B∈L(X).

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$ fixed positive integers. If $R$ admits a generalized derivation $F$ associated with a nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for all $x,y\in I$, then $R$ is commutative. Moreover we also examine the case when $R$ is a semiprime ring.

Let R be a prime ring, extended centroid C, Utumi quotient ring U, and m, n ≥ 1 are fixed positive integers, F a generalized derivation associated with a nonzero derivation d of R. We study the case when one of the following holds: (i) F(x)∘md(y)=(x∘y)n and (ii) (F(x)∘d(y))m=(x∘y)n, for all x, y in some appropriate subset of R. We also examine the case where R is a semiprime ring.

Let R be a prime ring of char(R)≠2, Z the center of R, and L a nonzero Lie ideal of R. If R admits a generalized derivation F associated with a derivation d which acts as a homomorphism or as anti-homomorphism on L, then either d=0 or L⊆Z. This result generalizes a theorem of Wang and You.

Let $R$ and $S$ be rings of a semi-projective Morita context, and $\alpha, \beta$ be automorphisms of $R$. An additive mapping $F$: $R\to R$ is called a generalized $(\alpha,\beta)$-derivation on $R$ if there exists an $(\alpha,\beta)$-derivation $d$: $R\to R$ such that $F(xy)=F(x)\alpha(y)+\beta(x)d(y)$ holds for all $x,y \in R$. For any $x,y \in R$, set $[x, y]_{\alpha, \beta} = x \alpha(y) - \beta(y) x$ and $(x \circ y)_{\alpha, \beta} = x \alpha(y) + \beta(y) x$. In the present paper, we shall show that if the ring $S$ is reduced then it is a commutative, in a compatible way with the ring $R$ . Also, we obtain some results on bialgebras via Cauchy modules.

Let R be a prime ring with characteristic different from 2 and L be a Lie ideal of R. In this paper, we characterize generalized left derivation, which acts as a homomorphisms or an anti-homomorphisms on L.

Let n ⩾ 1 be a fixed integer and let R be an (n + 1)!-torsion free ∗-ring with identity element e. If F, d:R → R are two additive mappings satisfying F(xn+1) = F(x)(x∗)n + xd(x)(x∗)n−1 + x2d(x)(x∗)n−2+ ⋯ +xnd(x) for all x ∈ R, then d is a Jordan ∗-derivation and F is a generalized Jordan ∗-derivation on R.

Let R be a ring and α, β be automorphisms of R. An additive mappingF: R → R is called a generalized (α,β)-derivation on R if there exists an (α,β)-derivation d:R → R such that F(xy)=F(x)α(y) + β(x(y) holds for all x, y ∈ R. For any x, y ∈ R, set [x, y]_{α,β} = xα(y)−(y)x and (x◦y)_{α,β} = xα(y)+β(y)x. In the present paper, we shall discuss the commutativity of a prime ring R admitting generalized (α,β)-derivations F and G satisfying any one of the following properties:(i) F([x, y]) = (x◦y)_{α,β}, (ii) F(x◦y) = [x,y]α,β, (iii) [F(x),y]_{α,β}=(F(x)◦y)_{α,β}, (iv) F([x, y]) = [F(x), y]_{α,β}, (v) F(x◦y) = (F(x◦y)_{α,β}, (vi) F([x,y]=[α(x),G(y)] and (vii) F(x◦y)=(α(x)◦G(y)) for all x, y in some appropriatesubset of R. Finally, obtain some results on semi-projective Morita context with generalized (α, β)-derivations.

Let R be a 2-torsion free ring and let L be a noncentral Lie ideal of R, and let F:R→R and G:R→R be two generalized derivations of R. We will analyse the structure of R in the following cases: (a) R is prime and F(um)=G(un) for all u∈L and fixed positive integers m≠n; (b) R is prime and F((upvq)m)=G((vrus)n) for all u,v∈L and fixed integers m,n,p,q,r,s≥1; (c) R is semiprime and F((uv)n)=G((vu)n) for all u,v∈[R,R] and fixed integer n≥1; and (d) R is semiprime and F((uv)n)=G((vu)n) for all u,v∈R and fixed integer n≥1.