Radical (mathematics)

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

In ring theory

Prime radical

Let it be a ring with a single element. The intersection over all prime ideals of is called the prime radical of . It is the smallest semi-prime ideal and a Nile ideal . ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$

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 ${\displaystyle R}$ ${\displaystyle {\mathfrak {a}}\subset R}$ ${\displaystyle R}$

${\displaystyle {\sqrt {\mathfrak {a}}}:=\{x\in R\mid \exists r\in \mathbb {N} :x^{r}\in {\mathfrak {a}}\}}$

the radical of . This is sometimes also referred to as or with . It's an ideal in . ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle r({\mathfrak {a}})}$ ${\displaystyle {\mathfrak {r}}({\mathfrak {a}})}$ ${\displaystyle R}$

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 . ${\displaystyle {\sqrt {(0)}}}$ ${\displaystyle \operatorname {nil} (R)}$ ${\displaystyle {\mathfrak {N}}_{R}}$ ${\displaystyle {\mathfrak {n}}_{R}}$

Jacobson radical

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. ${\displaystyle x-a}$ ${\displaystyle a}$

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 different prime factors ; this is a multiplicative function :

${\displaystyle \displaystyle \mathrm {rad} (n)=\prod _{p\mid n \atop p{\text{ prim}}}p}$

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

${\displaystyle \mathrm {rad} (324)=\mathrm {rad} (2^{2}\cdot 3^{4})=2\cdot 3=6}$.

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

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

The radical of a Lie group is the largest connected , resolvable normal divisor .

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

The radical of a square set, or more specifically a projective quadric, is the set of points in this set or quadric in which the tangent space consists of all points of the total space.

literature

• 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 .
• geometry
  • Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry . From the basics to the applications (=  Vieweg Studium: advanced course in mathematics ). 2nd, revised and expanded edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X ( table of contents [accessed April 1, 2012]).

Individual evidence

1. ^ Atiyah: Introduction To Commutative Algebra. 1969 p. 8
2. ^ Isaacs: Algebra, a graduate course. P. 420
3. ^ Atiyah: Introduction To Commutative Algebra. 1969 p. 5