Radical of an ideal

In the mathematical discipline of algebra, there are different meanings of the word radical.

In ring theory

Primradikal

It is a ring with unit element. The average over all prime ideals of means of Primradikal. It is the smallest Semiprimideal and a nil ideal.

In the case of a commutative ring, it coincides with the nilradical (see below).

Commutative case: a radical ideal and nilradical

It is a commutative ring with and an ideal in. Then one calls with

The radical of. Partly this is also referred to with or with. It is an ideal in.

An ideal which is identical to its radical, called radical ideal. Each Semiprimideal is a radical ideal.

The nilradical or nilpotent radical of a ring R, ie the set of nilpotent elements of the ring. Sometimes it is also referred to with or with or with. It is equal to the Primradikal, ie the intersection of all prime ideals. If the nilradical the zero ideal, that is, is zero the only nilpotent element, this means reducing the ring.

Jacobson radical

The intersection of all maximal left ideals of a ring is called the Jacobson radical.

Resolution of a polynomial by radicals

In the Galois theory, one deals with the resolution of polynomials into radicals, ie factors, a term used to describe that must be represented only by rational numbers, using the four basic arithmetic operations and using roots.

In group theory

The radical of a group is the largest solvable normal subgroup.

In number theory

The radical of an integer is the product of its prime factors different; this is a multiplicative function:

The radical of a prime number is the prime number itself, since the same prime factors are counted only once, all the powers of a number have the same radical.

Example: The number 324 has the radical 6 because

The radicals of the first natural numbers are as follows: 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13 ... ( in sequence A007947 OEIS )

An important role play radicals in the abc- conjecture.

In the theory of Lie algebras

The radical of a ( finite-dimensional ) Lie algebra is the largest solvable ideal.

In projective geometry

The radical of a quadratic quantity or special projective quadric is the set of points of this quantity or quadric, where there is the tangent space from all points of the overall space.