In ring theory
In the case of a commutative ring, it agrees with the Nile radical (see below).
Commutative case: radical of an ideal and Nile radical
Let it be a commutative ring with 1 and an ideal in . Then one designates with
the radical of . This is sometimes also referred to as or with . It's an ideal in .
An ideal that is identical to its radical is called a radical ideal . Every semi-prime ideal is a radical ideal. Radicals and radical ideals play an important role in algebraic geometry , they appear in Hilbert's zero theorem .
The Nile radical or nilpotent radical of a ring is R , i.e. the amount of nilpotent elements in the ring. Sometimes it is also referred to as or with or with . It is the same as the prime radical, i.e. the intersection of all prime ideals. Is the Nile radical the null ideal, i. H. if zero is the only nilpotent element, the ring is called reduced .
The intersection of all maximum left ideals of a ring is called a Jacobson radical .
Resolution of a polynomial by radicals
The Galois theory deals with the resolution of polynomials into radicals, i.e. into factors , describing an expression that only has to be representable by rational numbers, by means of the four basic arithmetic operations and by using roots.
In group theory
In number theory
The radical of a prime number is the prime number itself. Since the same prime factors are counted only once, all powers of a number have the same radical.
Example: The number 324 has the radical 6, da
Radicals play an important role in the abc conjecture .
In the theory of Lie algebras
The radical of a (finite-dimensional) Lie algebra is the greatest solvable ideal.
In Lie group theory
In the theory of algebraic groups
The unipotent radical of an algebraic group is a maximally closed, connected and unipotent normal divisor.
In projective geometry
- Ring theory:
- MF Atiyah ; IG Macdonald : Introduction to Commutative Algebra . Oxford 1969, Addison-Wesley Publishing Company, ISBN 0-201-00361-9 .
- Martin Isaacs: Algebra, a graduate course . 1st edition. Brooks / Cole Publishing Company, 1993, ISBN 0-534-19002-2 .
- Hideyuki Matsumura: Commutative ring theory . 2nd Edition. Cambridge University Press , 1989, ISBN 978-0-521-36764-6 .
- Atiyah: Introduction To Commutative Algebra. 1969 p. 8
- Isaacs: Algebra, a graduate course. P. 420
- Atiyah: Introduction To Commutative Algebra. 1969 p. 5