Radical of an ideal

The radical of an ideal is the smallest ideal that contains all the nth roots of elements in the original ideal, where n is a positive integer. It is computed by taking the intersection of all prime ideals that contain the original ideal. The radical is useful in simplifying calculations and understanding the properties of the variety associated with the ideal.From: Near Rings, Fuzzy Ideals, and Graph Theory [2019], Handbook of Graphical Models [2018]

Algebraic Aspects of Conditional Independence and Graphical Models

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Published in Marloes Maathuis, Mathias Drton, Steffen Lauritzen, Martin Wainwright, Handbook of Graphical Models, 2018

Thomas Kahle, Johannes Rauh, Seth Sullivant

Radical ideals can also be characterized algebraically, and there are algorithms to compute radicals. The radical is usually a simpler ideal, and if the radical of an ideal can be computed, it is advantageous to do this in a first step in each calculation (as long as one is only interested in properties of V(J), and not in algebraic properties of J).

On the finiteness of accessibility test for nonlinear discrete-time systems

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Published in International Journal of Control, 2021

Mohammad Amin Sarafrazi, Ewa Pawłuszewicz, Zbigniew Bartosiewicz, Ülle Kotta

Let us recall some basic facts from ideal theory. An idealI of a commutative ring is a subset of with the properties that if and then and . The radical of an ideal I of , denoted by , is the set . If I coincides with its own radical, then I is called a radical ideal. The real radical of I, denoted by , is the set of all for which there are natural m, k and such that . If I, J are ideals of , then (i) the real radical of I is an ideal of , (ii) , (iii) if then , see Bochnak, Coste, and Roy (1998). A semialgebraic (respectively semianalytic) set X is a set such that for every there is an open neighbourhood V of x with property that is a finite Boolean combination of sets and where are polynomial functions (respectively analytic functions). For a set , the Zariski closure of A is defined as the smallest algebraic variety containing A, and is denoted by .

Radical of an ideal

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