a branch of mathematics, the radical of an ideal I {\displaystyle I} of a commutative ring is another ideal defined by the property that an element x...

branch of mathematics, a radical of a ring is an ideal of "not-good"[definition needed] elements of the ring. The first example of a radical was the...

mathematics, the radical symbol, radical sign, root symbol, or surd is a symbol for the square root or higher-order root of a number. The square root of a number...

mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right...

In algebra, the real radical of an ideal I in a polynomial ring with real coefficients is the largest ideal containing I with the same (real) vanishing...

Radical of an ideal, an important concept in abstract algebra Radical of a ring, an ideal of "bad" elements of a ring Jacobson radical, consisting of...

specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even...

over a field, then the radical of an ideal in A {\displaystyle A} is the intersection of all maximal ideals containing the ideal (because A {\displaystyle...

ideal in it is maximal. The notion of a Goldman ideal can be used to give a slightly sharpened characterization of a radical of an ideal: the radical...

I(V(J))={\sqrt {J}}.} Radical ideals (ideals that are their own radical) of k[V] correspond to algebraic subsets of V. Indeed, for radical ideals I and J, I ⊆...

multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary and semiprime. An ideal P of a...

maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R...

(ACC) and descending chain condition (DCC) Fractional ideal Ideal class group Radical of an ideal Hilbert's Nullstellensatz Flat module Flat map Flat map...

the radical of an ideal, the ideal of zero-dimensional schemes, Poincaré series and Hilbert functions, factorization of polynomials, and toric ideals. The...

and radicals; e.g., if I {\displaystyle {\sqrt {I}}} denote the radical of an ideal I in R, then I ⋅ S − 1 R = I ⋅ S − 1 R . {\displaystyle {\sqrt {I}}\cdot...

closure of an integral domain in a field that contains it. The radical of an ideal in a commutative ring. In geometry, the convex hull of a set S of points...

ideal is an ideal consisting of nilpotent elements. 2.  The (Baer) upper nil radical is the sum of all nil ideals. 3.  The (Baer) lower nil radical is...

ideal Principal ideal Ideal quotient Maximal ideal, minimal ideal Primitive ideal, prime ideal, semiprime ideal Radical of an ideal Jacobson radical Socle...

non-zero abelian ideals; g {\displaystyle {\mathfrak {g}}} has no non-zero solvable ideals; the radical (maximal solvable ideal) of g {\displaystyle {\mathfrak...

ring is the set of all nilpotent elements in the ring, or equivalently the radical of the zero ideal. This is an ideal because the sum of any two nilpotent...

and this ideal is called the associated prime ideal of Q. In this situation, Q is said to be P-primary. On the other hand, an ideal whose radical is prime...

{cont} (fg)}}} where ⋅ {\displaystyle {\sqrt {\cdot }}} denotes the radical of an ideal. Moreover, if R {\displaystyle R} is a GCD domain (e.g., a unique...

{\displaystyle k=K} we obtain an order-reversing bijective correspondence between the algebraic sets in Kn and the radical ideals of K [ X 1 , … , X n ] . {\displaystyle...

In mathematics, more specifically ring theory, an ideal I of a ring R is said to be a nilpotent ideal if there exists a natural number k such that I k...

  A radical of an element x of a ring is an element such that some positive power is x. 4.  The radical of an ideal is the ideal of radicals of its elements...

generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the irrelevant ideal. The terminology arises from...

maximal ideal and thus ( x ) = p {\displaystyle {\sqrt {(x)}}={\mathfrak {p}}} . Since ( x ) {\displaystyle (x)} contains some power of its radical, it follows...

domain of Krull dimension n. An analog of Hilbert's Nullstellensatz is valid: the radical of an ideal is the intersection of all maximal ideals containing...

{\displaystyle a_{i}\in I^{i}} with r {\displaystyle r} as a root. The radical of an ideal is integrally closed. For noetherian rings, there are alternate definitions...

generalization of a commutative ring Divisibility (ring theory): nilpotent element, (ex. dual numbers) Ideals and modules: Radical of an ideal, Morita equivalence...