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Our fundamental aim in this paper is to study certain * {*} -identities with pair of derivations acting on a prime ideal P and investigate commutativity of a quotient ring R / P {R/P} where R is any ring. Some well-known results characterizing commutativity of prime rings have been generalized. Finally, examples are given to demonstrate that the restrictions imposed on the hypothesis of our results are not superfluous. [ABSTRACT FROM AUTHOR]

The purpose of this article is to enhance the previous studies regarding the behavior of a quotient ring ℑ / ℘ , where ℘ is a prime ideal in a ring ℑ. In particular, we are going to explore more general scenarios whenever a ring ℑ admits a generalized (α , β) -derivation associated with an (α , β) -derivation ∂ that satisfies certain criteria involving ℘, where α and β are automorphisms on ℑ. Moreover, we provide some examples to demonstrate the importance of the assumptions made in our results. [ABSTRACT FROM AUTHOR]

The purpose of this article is to enhance the previous studies regarding the behavior of a quotient ring ℑ/℘, where ℘ is a prime ideal in a ring ℑ. In particular, we are going to explore more general scenarios whenever a ring ℑ admits a generalized (α,β)-derivation associated with an (α,β)-derivation ∂ that satisfies certain criteria involving ℘, where α and β are automorphisms on ℑ. Moreover, we provide some examples to demonstrate the importance of the assumptions made in our results.

The q-rung orthopair fuzzy atmosphere is an innovative approach for handling unclear circumstances in a range of decision making problems. As compare to intuitionistic fuzzy sets, this one is more appropriate and adaptable because it evaluates the significance of ring theory while retaining the features of q-rung orthopair fuzzy sets. In this study, we characterize q-rung orthopair fuzzy subring as a modification of the pythagorean fuzzy subring. We introduce the novel idea of q-rung orthopair fuzzy subring and investigate the algebraic characteristics for the q-rung orthopair fuzzy subrings. Furthermore, we establish the concept of q-rung orthopair fuzzy quotient ring and q-rung orthopair fuzzy left and right ideals. Also, we describe the q-rung orthopair fuzzy level subring and associate axioms. Finally, we investigate how ring homomorphism influences the q-rung orthopair fuzzy subring and investigate there pre-images homomorphism on q-ROFSR and different aspects of images.

A ring R is called an FMR-ring if it is finite modulo all its maximal ideals. In this paper, we investigate some properties of FMR-rings in various ring extensions. Mainly we are concerned with the charachterization of FMR-rings arising from bi-amalgamated algebras and trivial ring extensions. [ABSTRACT FROM AUTHOR]

The q-rung orthopair fuzzy atmosphere is an innovative approach for handling unclear circumstances in a range of decision making problems. As compare to intuitionistic fuzzy sets, this one is more appropriate and adaptable because it evaluates the significance of ring theory while retaining the features of q-rung orthopair fuzzy sets. In this study, we characterize q-rung orthopair fuzzy subring as a modification of the pythagorean fuzzy subring. We introduce the novel idea of q-rung orthopair fuzzy subring and investigate the algebraic characteristics for the q-rung orthopair fuzzy subrings. Furthermore, we establish the concept of q-rung orthopair fuzzy quotient ring and q-rung orthopair fuzzy left and right ideals. Also, we describe the q-rung orthopair fuzzy level subring and associate axioms. Finally, we investigate how ring homomorphism influences the q-rung orthopair fuzzy subring and investigate there pre-images homomorphism on q-ROFSR and different aspects of images. [ABSTRACT FROM AUTHOR]

Let ℚ be the rational filed. For a square-free integer d with d ≠ 0 , 1 , we denote by K = ℚ (d) the quadratic field. Let K be the ring of algebraic integers of K. In this paper, we completely determine the unit group of the quotient ring K / 〈 ξ n 〉 of K for an arbitrary prime ξ in ℚ (d) , where ℚ (d) has the unique factorization property, and n > 1 is a rational integer. [ABSTRACT FROM AUTHOR]

In this paper we investigate the following result. Let R be a prime ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R. If F and G are the two generalized derivation of R such that (F(xy) + G(yx))n − (xy ∓ yx)n = 0, for all x, y ∈ I, then either R is commutative or F (x) = x, G(x) = ∓x for all x ∈ R and n = 1.

The main purpose of this paper is to develop a new approach that consists to investigate the structure of a quotient ring R/P via action of generalized derivations on the prime ideal P. In this direction, we initiate new classes of additive mappings extending both of centralizing and commuting mappings. Furthermore, for an arbitrary ring, we will consider algebraic identities based on its prime ideals. As an application, we obtain some results concerning invariance of minimal prime ideals of a semiprime ring under derivations. [ABSTRACT FROM AUTHOR]

We study the right duo property on regular elements, and we say that rings with this property are right DR. It is first shown that the right duo property is preserved by right quotient rings when the given rings are right DR. We prove that the polynomial ring over a ring R is right DR if and only if R is commutative. It is also proved that for a prime number p , the group ring K G of a finite p -group G over a field K of characteristic p is right DR if and only if it is right duo, and that there exists a group ring K G that is neither DR nor duo when G is not a p -group. [ABSTRACT FROM AUTHOR]

The fundamental aim of this paper is to investigate the structure of a quotient ring R/P where R is an arbitrary ring and P is a prime ideal of R. More precisely, we will characterize the commutativity of R/P via the behavior of generalized derivations of R satisfying algebraic identities involving the prime ideal P. Moreover, various well-known results characterizing the commutativity of prime (semi-prime) rings have been extended. Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous. [ABSTRACT FROM AUTHOR]

We introduce a general framework encompassing the main hard problems emerging in lattice-based cryptography, which naturally includes the recently proposed Mersenne prime cryptosystem, but also problems coming from code-based cryptography. The framework allows to easily instantiate new hard problems and to automatically construct plausibly post-quantum secure primitives from them. As a first basic application, we introduce two new hard problems and the corresponding encryption schemes. Concretely, we study generalisations of hard problems such as SIS, LWE and NTRU to free modules over quotients of ℤ[X] by ideals of the form (f, g), where f is a monic polynomial and g ∈ ℤ[X] is a ciphertext modulus coprime to f. For trivial modules (i.e. of rank one), the case f = Xn + 1 and g = q ∈ ℤ>1 corresponds to ring-LWE, ring-SIS and NTRU, while the choices f = Xn – 1 and g = X – 2 essentially cover the recently proposed Mersenne prime cryptosystems. At the other extreme, when considering modules of large rank and letting deg(f) = 1, one recovers the framework of LWE and SIS.

Let R be a ring and σ be an endomorphism of R. Recall that a ring R is said to be a left σ-(S, 1) ring if for a, b ∈ R, ab = 0 implies that aRb = 0 and σ(a)Rb = 0. In this paper we discuss a stronger type of primary decomposition (known as transparency) of a left σ-(S, 1) ring R, and Ore extension R[x; σ]. [ABSTRACT FROM AUTHOR]

In this article, we introduce the notions of normed quotient ring, normed quotient subring, normed quotient ring homomorphism, normed quotient ring natural homomorphism and investigate some of their related properties. [ABSTRACT FROM AUTHOR]