Related Experiment Video
Updated: Apr 17, 2026
06:42
Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
9.8K
Generalized higher (U,M)-derivations in prime Γ-Rings.
Md Mizanor Rahman1, Akhil Chandra Paul2
1Department of Mathematics, Jagannath University, Dhaka, Bangladesh.
Springerplus
|February 13, 2015
Summary
This study investigates generalized higher (U,M)-derivations on prime Γ-rings. We establish a key result concerning the behavior of these derivations under specific ring conditions.
Area of Science:
- Abstract algebra
- Ring theory
- Non-associative algebra
Background:
- Prime Γ-rings are generalizations of prime rings with additional structure.
- Lie ideals play a crucial role in understanding the algebraic properties of rings.
- Generalized higher derivations extend the concept of derivations in ring theory.
Purpose of the Study:
- To explore the properties of generalized higher (U,M)-derivations in a specific class of Γ-rings.
- To establish a new identity involving these derivations and Lie ideals.
- To contribute to the understanding of algebraic structures in non-associative settings.
Main Methods:
- Utilizing the properties of 2-torsion free prime Γ-rings.
- Applying the definition of admissible Lie ideals.
- Leveraging the concept of generalized higher derivations and associated higher derivations.
- Employing algebraic manipulations and ring identities.
Main Results:
- A specific identity involving a generalized higher (U,M)-derivation F and an associated higher (U,M)-derivation D on a prime Γ-ring M is proven.
- The identity holds for all elements within the admissible Lie ideal U and for all n in N.
- The condition a α b β c = a β b α c is fundamental to the derivation of the result.
Conclusions:
- The established identity provides new insights into the structure of generalized higher derivations in prime Γ-rings.
- The findings extend existing results in the theory of derivations and Γ-rings.
- This research deepens the understanding of algebraic mappings in abstract algebraic structures.
Related Concept Videos
01:29Higher Derivatives
302
In calculus, higher-order derivatives extend the idea of differentiation beyond the first derivative to capture successive rates of change. These derivatives provide detailed information about the behavior of functions and have important applications in both mathematics and physics. To illustrate these concepts, consider the example function\begin{equation*}f(x) = x^3 - x\end{equation*}which serves as a useful case study for exploring higher derivatives.The first derivative represents the slope...
302
01:25Summation Notation
355
Sigma notation, also known as summation notation, provides a concise method for representing the sum of a sequence of terms that follow a regular pattern. It utilizes the uppercase Greek letter sigma (∑), A typical expression is:In this form, k the index of summation is 1, the starting value, and n the ending value. The term ak represents the general term of the sequence.For example, the increasing sequence 5, 7, 9, ..., 23 over 10 terms can be expressed as:This simplifies the...
355
01:29Indeterminate Products
132
Indeterminate forms also arise in the evaluation of limits involving products, particularly when one factor approaches zero while the other tends to positive or negative infinity. This situation, commonly described as a zero-times-infinity form, does not have an immediately interpretable outcome. Depending on how the factors behave relative to one another, the limit of such a product may be zero, infinite, or a finite nonzero value.Product Limits and Algebraic RewritingTo analyze limits of this...
132
01:30Fundamental Theorem of Algebra
459
The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as: with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the...
459
01:15Determination of Pi Terms
752
The Buckingham Pi theorem is a valuable method in dimensional analysis, reducing complex relationships between variables into dimensionless terms. Relevant variables in analyzing the lift force on an airplane wing include lift force, air density, wing area, aircraft velocity, and air viscosity. Expressing each variable in terms of fundamental dimensions — mass, length, and time — provides a consistent foundation for constructing these dimensionless terms.
The theorem indicates that...
The theorem indicates that...
752
01:24Substitution Rule Applied to Definite Integrals
222
When evaluating a definite integral whose integrand matches the structure of a composite function, the substitution method provides an efficient way to simplify the calculation. This method is based on reversing the chain rule from differentiation, allowing a complicated expression to be rewritten in a simpler form. When the integrand contains an inner function and its derivative, substitution naturally reduces the complexity of the problem.The core idea of substitution for definite integrals...
222