Search Results (39)

This paper studies structural properties of semiprimes $N=pq$ in computational number theory, focusing on cases where the prime factors are close. We analyze the relationship between $\sqrt{N}$ and $\sqrt{\phi \left(N\right)}$ and show that, under a bounded prime gap condition, these quantities exhibit strong proximity. Specifically, assuming $|p-q|\le 2^{l/4}$ for an l-bit semiprime, we prove that the Euler totient function admits the exact representation $\phi \left(N\right)=N-1-2\lfloor \sqrt{N}\rfloor$. Based on this result, we develop an interval-based method for reconstructing $\phi \left(N\right)$ within a narrow neighborhood derived from square-root bounds, followed by a discriminant-based refinement step for recovering the prime factors. Experimental evaluation on large semiprimes, including RSA-type moduli of 4095 and 4096 bits, shows that the method operates efficiently under the stated structural condition using only elementary integer arithmetic. These results provide a theoretical characterization of semiprimes with small prime gaps and offer a framework for identifying structurally weak RSA moduli. This method, given its high efficiency when the prime factors are close to each other, can be regarded as an alternative to Fermat’s factorization method. In particular, for semiprime integers with a small prime gap (i.e., $|p-q|$ is small), the proposed approach exploits structural properties based on the proximity of square roots, thereby significantly accelerating the factorization process. Consequently, it not only aligns with the theoretical foundation of Fermat’s method but, under certain conditions, may also achieve comparable or even superior practical performance. Full article

In this paper, we investigate the stability and superstability of a specific class of functional inequalities associated with centrally extended ∗-derivations on Banach ∗-algebras. A CE ∗-derivation $\delta :\mathcal{R}\to \mathcal{R}$ is defined as an additive mapping satisfying $\delta (x+y)-\delta \left(x\right)-\delta \left(y\right)\in Z\left(\mathcal{R}\right)$ and $\delta \left(xy\right)-\delta \left(x\right)y^{\ast }-x\delta \left(y\right)\in Z\left(\mathcal{R}\right)$ for all $x,y\in \mathcal{R}$, where $Z\left(\mathcal{R}\right)$ denotes the center of the ring. We consider the functional inequality $\parallel [a_1\delta \left(x_1\right)+a_2\delta \left(x_2\right)+a_3\delta \left(x_3\right),w]\parallel \le \parallel [\delta (a_1{x}_1+a_2{x}_2+a_3{x}_3),w]\parallel + \mathrm{\Phi }(x_1,x_2,x_3,w),$ where $\mathrm{\Phi }$ is a perturbing term. By employing the direct method, we establish several theorems concerning the Hyers–Ulam stability of this inequality in the context of unital Banach ∗-algebras. Furthermore, we provide sufficient conditions under which these functional inequalities exhibit superstability. We also explore the implications of our results for linear ∗-derivations in semiprime Banach ∗-algebras with no nonzero central ideals. Full article

Let ℜ be a unital algebra over a field $\mathbb{k}$ with $char\left(\mathbb{k}\right)\ne 2$, and let $\mathsf{ϝ},\xi ,\zeta :\Re \to \Re$ be linear mappings. We say that $\mathsf{ϝ}$ is a $\{\xi ,\zeta \}$-derivation if $\mathsf{ϝ}\left(\vartheta \varsigma \right)=\xi \left(\vartheta \right)\varsigma +\vartheta \zeta \left(\varsigma \right)=\zeta \left(\vartheta \right)\varsigma +\vartheta \xi \left(\varsigma \right)\mathrm{for}\mathrm{all}\vartheta ,\varsigma \in \Re .$ The mapping $\mathsf{ϝ}$ is said to be a Lie $\{\xi ,\zeta \}$-derivation if $\mathsf{ϝ}\left(\right[\vartheta ,\varsigma \left]\right)=\left[\xi \right(\vartheta ),\varsigma ]+[\vartheta ,\zeta (\varsigma \left)\right]\mathrm{for}\mathrm{all}\vartheta ,\varsigma \in \Re ,$ where $[\vartheta ,\varsigma ]=\vartheta \varsigma -\varsigma \vartheta$ denotes the Lie product. In this paper, we prove that if every Lie $\{\xi ,\zeta \}$-derivation on ℜ is necessarily a $\{\xi ,\zeta \}$-derivation, then the same property holds for the tensor product algebra $\Re \otimes \Im$, where ℑ is any commutative unital algebra. Moreover, every Lie $\{\xi ,\zeta \}$-derivation of a semiprime algebra is a $\{\xi ,\zeta \}$-derivation. As a consequence, Lie derivations on tensor products of semiprime algebras with commutative algebras reduce to derivations in the classical sense. Full article

This work explores the notion of axis-reversible rings, a generalization of axis-commutative rings. The objective is to investigate their characteristics and relevance within the wider context of ring theory. This paper defines axis-reversibility and demonstrates its importance through many examples. It also analyzes the characteristics of several matrix rings, elucidating the conditions under which a ring can be deemed axis-reversible. This paper examines the relationship between axis-reversibility and other significant ring qualities, such as reducedness and semiprimeness, through comprehensive arguments and proofs. This study provides novel perspectives on non-commutative rings, enhancing our comprehension of algebraic structures. Full article

Let $\mathfrak{R}$ be a prime ring, and let F denote a generalized derivation associated with a derivation d of $\mathfrak{R}$. Consider I as a nonzero ideal of $\mathfrak{R}$, and let $m,n,k,l$ be fixed positive integers. In this study, we explore the behavior of the generalized derivation F within the structures of both prime and semiprime rings that satisfy the functional identity ${[F\left(\eta \right),d\left(\tau \right)]}_m={\eta }^n{[\eta ,\tau ]}_l{\eta }^k$, $\forall ,\eta ,\tau \in I.$ Furthermore, we extend this investigation to the framework of Banach algebras, analyzing how generalized derivations operate in such algebras. A comparative discussion is also presented to highlight the distinctions and similarities in the behavior of generalized derivations within Banach algebraic settings under the above structural condition. Full article

This paper introduces and examines the concept of weak 2-local inner derivations and their relationship to $\mathcal{P}$-2-local mappings. It is established that every such mapping on a semiprime ∗-Banach algebra with a faithful trace is a derivation, which also provides a complete characterization on finite von Neumann algebras. Additionally, it is shown that weak 2-local inner derivations coincide with the 2-local reflection closure of the inner derivations. Full article

This work establishes a unified structural theory for non-global Lie higher derivations on triangular algebras $\mathcal{T}$, without assuming the existence of a unit element. The primary contribution is the introduction of extreme non-global Lie higher derivations and the proof that every non-global Lie higher derivation on $\mathcal{T}$ admits a unique decomposition into three components: a higher derivation, an extreme non-global Lie higher derivation, and a central map vanishing on all commutators $[x,y]$, where $x,y\in \mathcal{T}$ satisfy $xy=0$. This general framework is then explicitly applied to describe such derivations on two significant classes of algebras: upper triangular matrix algebras over faithful algebras and over semiprime algebras. By encompassing both unital and non-unital cases within a single characterization, the theory developed here not only generalizes numerous earlier results but also substantially expands the scope of the existing research landscape. Full article

If M is an algebra in a semidegenerate congruence-modular variety $\mathcal{V}$, then the set $Con(M)$ of congruences of M is an integral complete l-groupoid (= $icl$-groupoid). For any morphism $f:M\to N$ of $\mathcal{V}$, consider the map $f^{•}:Con(M)\to Con(N)$, where, for each congruence $\epsilon$ of M, $f^{•}(\epsilon )$ is the congruence of N generated by $f(\epsilon )$. Then, $f^{•}$ is a semimorphism of $icl$-groupoids, i.e., it preserves the arbitrary joins and the top congruences. The neo-commutative $icl$-groupoids were introduced recently by the author as an abstraction of the lattices of congruences of Kaplansky neo-commutative rings. In this paper, we define the admissible semimorphisms of $icl$-groupoids. The basic construction of the paper is a covariant functor defined by the following: $(1)$ to each semiprime and neo-commutative $icl$-groupoid A, we assign a coherent frame $R(A)$ of radical elements of A; and $(2)$ to an admissible semimorphism of $icl$-groupoids $u:A\to B$, we assign a coherent frame morphism $u^{\rho }:R(A)\to R(B)$. By means of this functor, we transfer a significant amount of results from coherent frames and coherent frame morphisms to the neo-commutative $icl$-groupoids and their admissible semimorphisms. We study the m-prime spectra of neo-commutative $icl$-groupoids and the going-down property of admissible semimorphisms. Using some transfer properties, we characterize some classes of admissible semimorphisms of $icl$-groupoids: Baer and weak-Baer semimorphisms, quasi r-semimorphisms, quasi $r^{*}$-semimorphisms, quasi rigid semimorphisms, etc. Full article

Let S be a 2-torsion free semiprime ring and U be a noncentral square-closed Lie ideal of $S.$ An additive mapping ℏ on S is defined as a homoderivation if $\hslash \left(ab\right)=\hslash \left(a\right)\hslash \left(b\right)+\hslash \left(a\right)b+a\hslash \left(a\right)$ for all $a,b\in S.$ In the present paper, we shall prove that ℏ is a commuting map on U if any one of the following holds: (i)$\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)+{\tilde{a}}_1{\tilde{a}}_2\in Z,$(ii)$\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)-{\tilde{a}}_1{\tilde{a}}_2\in Z,$(iii)$\hslash \left({\tilde{a}}_1\circ {\tilde{a}}_2\right)=0,$(iv)$\hslash \left({\tilde{a}}_1\circ {\tilde{a}}_2\right)=\left[{\tilde{a}}_1,{\tilde{a}}_2\right],$(v)$\hslash \left(\left[{\tilde{a}}_1,{\tilde{a}}_2\right]\right)=0,$(vi)$\hslash \left(\left[{\tilde{a}}_1,{\tilde{a}}_2\right]\right)=$ $({\tilde{a}}_1\circ {\tilde{a}}_2),$(vii)${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_1{\tilde{a}}_2\in Z,$(viii)${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_2{\tilde{a}}_1=0,$(ix)${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_1\circ {\tilde{a}}_2=0,$(x)$[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]\pm {\tilde{a}}_1{\tilde{a}}_2=0,$(xi)$[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]\pm {\tilde{a}}_2{\tilde{a}}_1=0,$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in U,$ where ℏ is a homoderivation on $S.$ Full article

This paper aims to introduce a novel idea of possibility multi-fuzzy soft ordered semigroups for ideals and interior ideals. Various results, formulated as theorems based on these concepts, are presented and further validated with suitable examples. This paper also explores the broad applicability of possibility multi-fuzzy soft ordered semigroups in solving modern decision-making problems. Furthermore, this paper explores various classes of ordered semigroups, such as simple, regular, and intra-regular, using this innovative method. Based on these concepts, some important conclusions are drawn with supporting examples. Moreover, it defines the possibility of multi-fuzzy soft ideals for semiprime ordered semigroups. Full article

The current article focuses on studying the behavior of a ring $\Re /\Pi$ when ℜ admits generalized derivations $\Psi$ and $\Omega$ with associated derivations $\varphi$ and $\delta$, respectively. These derivations satisfy specific differential identities involving $\Pi$, where $\Pi$ is a prime ideal of an arbitrary ring ℜ, not necessarily prime or semiprime. Furthermore, we explore some consequences of our findings. To emphasize the necessity of the primeness of $\Pi$ in the hypotheses of our various theorems, we provide a list of examples. Full article

We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences with respect to the term condition commutator. Then we use the topological structure of the minimal prime spectrum to study extensions of universal algebras that generalize certain types of ring extensions. Our results hold for semiprime members of semidegenerate congruence–modular varieties, as well as semiprime algebras whose term condition commutators are commutative and distributive with respect to arbitrary joins and satisfy certain conditions on compact congruences, even if those algebras do not generate congruence–modular varieties. Full article

This paper focuses on examining a new type of n-additive map called the symmetric reverse n-derivation. As implied by its name, it combines the ideas of n-additive maps and reverse derivations, with a 1-reverse derivation being the ordinary reverse derivation. We explore several findings that expand our knowledge of these maps, particularly their presence in semiprime rings and the way rings respond to specific functional identities involving elements of ideals. Also, we provide examples to help clarify the concept of symmetric reverse n-derivations. This study aims to deepen our understanding of these symmetric maps and their properties within mathematical structures. Full article

This paper investigates the relationship between the commutativity of rings and the properties of their multiplicative generalized derivations. Let $\mathcal{F}$ be a ring with a semiprime ideal $\Pi$. A map $\varphi :\mathcal{F}\to \mathcal{F}$ is classified as a multiplicative generalized derivation if there exists a map $\sigma :\mathcal{F}\to \mathcal{F}$ such that $\varphi \left(xy\right)=\varphi \left(x\right)y+x\sigma \left(y\right)$ for all $x,y\in \mathcal{F}$. This study focuses on semiprime ideals $\Pi$ that admit multiplicative generalized derivations $\varphi$ and G that satisfy certain differential identities within $\mathcal{F}$. 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. Full article