Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Partial Fractions

Partial Fractions

A partial fraction is a component of a rational expression represented as the sum of simpler fractions. When a rational function is expressed as a ratio of two polynomials, it can often be decomposed into a sum of fractions whose denominators are simpler polynomials, typically linear or irreducible quadratic factors. This process is called partial fraction decomposition, and it is used to simplify complex expressions for integration, solving equations, or analysis.Partial fraction decomposition...

Radicals

Radicals

Roots, often written as radicals, identify the quantity that must be raised to a specific exponent to produce a given value. A radical expression consists of two main components: the radicand, which is the value placed inside the root symbol, and the index, which indicates the degree of the root being taken. The notation n√a indicates the principal nth root of a. If n equals 2, the operation is the square root, while n = 3 defines the cube root. When n is even, a negative radicand does...

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

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...

Radical Equations

Radical Equations

Radical equations are mathematical expressions in which the variable is found within a radical, most commonly a square root or cube root. These equations frequently arise in science, engineering, and real-world measurements involving nonlinear relationships. To solve a radical equation, the standard procedure is to isolate the radical expression and then eliminate the radical by raising each side to a power equal to the index of the radical. This process may lead to extraneous...

Radical Formation: Addition

Radical Formation: Addition

Radicals can be formed by adding a radical to a spin-paired molecule. This is typically observed with unsaturated species, where the addition of a radical across the π bond leads to the production of a new radical by dissolving the π bond. For example, the addition of a Br radical to an alkene yields a carbon-centered radical.
Similar to charge conservation in chemical reactions, spin conservation is implicit for radical reactions. Accordingly, the product formed must possess an...

Rational Expressions

Rational Expressions

Rational expressions are algebraic fractions in which both the numerator and the denominator are polynomials. These expressions follow the arithmetic rules of numerical fractions but require extra care due to the presence of variables. A fundamental part of working with rational expressions is identifying values that make the expression undefined, typically those that result in division by zero or undefined radicals.Determining the DomainThe domain of a rational expression includes all real...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same authorSame journal

On the arithmetic of stable domains.

Communications in algebra·2021

See all related articles

Search research articles

Related Experiment Video

Updated: Dec 24, 2025

Isolating Free Carbenes, their Mixed Dimers and Organic Radicals

Isolating Free Carbenes, their Mixed Dimers and Organic Radicals

Published on: April 19, 2019

Radical factorization in finitary ideal systems.

Bruce Olberding¹, Andreas Reinhart¹

¹Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, USA.

Communications in Algebra

|April 8, 2020

Summary

This summary is machine-generated.

This study explores radical factorization in cancellative monoids, introducing new characterizations for specific monoid types. It reveals conditions for monoid rings and *-Nagata rings to be SP-domains, advancing ideal theory.

Keywords:

13A15 13F05 20M12 20M13 Radical factorization ideal system modularization monoid ring

More Related Videos

Exploring the Radical Nature of a Carbon Surface by Electron Paramagnetic Resonance and a Calibrated Gas Flow

Exploring the Radical Nature of a Carbon Surface by Electron Paramagnetic Resonance and a Calibrated Gas Flow

Published on: April 24, 2014

Application of Elemental Lanthanides in the Selective C-F Activation of Trifluoromethylated Benzofulvenes Providing Access to Various Difluoroalkenes

Application of Elemental Lanthanides in the Selective C-F Activation of Trifluoromethylated Benzofulvenes Providing Access to Various Difluoroalkenes

Published on: July 28, 2018

Related Experiment Videos

Last Updated: Dec 24, 2025

Isolating Free Carbenes, their Mixed Dimers and Organic Radicals

Isolating Free Carbenes, their Mixed Dimers and Organic Radicals

Published on: April 19, 2019

Exploring the Radical Nature of a Carbon Surface by Electron Paramagnetic Resonance and a Calibrated Gas Flow

Exploring the Radical Nature of a Carbon Surface by Electron Paramagnetic Resonance and a Calibrated Gas Flow

Published on: April 24, 2014

Application of Elemental Lanthanides in the Selective C-F Activation of Trifluoromethylated Benzofulvenes Providing Access to Various Difluoroalkenes

Application of Elemental Lanthanides in the Selective C-F Activation of Trifluoromethylated Benzofulvenes Providing Access to Various Difluoroalkenes

Published on: July 28, 2018

Area of Science:

Abstract Algebra
Commutative Algebra
Ideal Theory

Background:

Investigates radical factorization within finitary ideal systems of cancellative monoids.
Builds upon existing research in algebraic structures and ideal systems.

Purpose of the Study:

To present novel characterizations for r-almost Dedekind r-SP-monoids.
To provide detailed descriptions of t-almost Dedekind t-SP-monoids and w-SP-monoids.
To establish conditions for monoid rings and *-Nagata rings to exhibit SP-domain properties.

Main Methods:

Utilizes concepts of radical factorization and ideal systems.
Employs characterization techniques for specific algebraic structures.
Analyzes properties of monoid rings and *-Nagata rings.

Main Results:

New characterizations for r-almost Dedekind r-SP-monoids are established.
Specific descriptions for t-almost Dedekind t-SP-monoids and w-SP-monoids are provided.
A key finding shows a monoid is a w-SP-monoid iff the radical of any nontrivial principal ideal is t-invertible.

Conclusions:

The study advances the understanding of radical factorization in cancellative monoids.
It offers criteria for identifying specific types of SP-monoids and related ring structures.
The findings contribute to the broader theory of ideals in abstract algebra.