Jacobson radical

In ring theory , a branch of algebra , the Jacobson radical of a ring denotes an ideal of that contains elements of that can be regarded as "close to zero". The Jacobson radical is named after Nathan Jacobson , who first studied it. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$

Jacobson radical of R modules

The following is a ring with one and an R-left module . ${\displaystyle R}$ ${\displaystyle M}$

definition

The average of all maximum submodules of is called the (Jacobson) radical (or short ). ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle \mathrm {Rad} _{R}(M)}$ ${\displaystyle \mathrm {Rad} (M)}$

Is finitely generated, then: . An element of is called superfluous if the following applies to each sub-module : From already follows . ${\displaystyle M}$ ${\displaystyle \mathrm {Rad} (M)=\{x\in M|x\ \mathrm {ist\ {\ddot {u}}berfl{\ddot {u}}ssig\ in} \ M\}}$ ${\displaystyle x}$ ${\displaystyle M}$ ${\displaystyle N\subset M}$ ${\displaystyle M=N+Rx}$ ${\displaystyle M=N}$

properties

• If it is finally generated and a sub-module of with , then it is already . This property is also known as the Nakayama lemma . ${\displaystyle M}$ ${\displaystyle N\subset M}$ ${\displaystyle M}$ ${\displaystyle M=N+\mathrm {Rad} (M)}$ ${\displaystyle M=N}$
• Is finally generated and then is . (This is the special case of the previous statement.) ${\displaystyle M}$ ${\displaystyle M\not =0}$ ${\displaystyle \mathrm {Rad} (M)\not =M}$ ${\displaystyle N=0}$
• ${\displaystyle \mathrm {Rad} (M)=0}$if and only if isomorphic easier to a sub-module of a direct product is -modules. ${\displaystyle M}$ ${\displaystyle R}$
• ${\displaystyle M}$is finitely generated and semi-simple if and only if artinian and is. ${\displaystyle M}$ ${\displaystyle \mathrm {Rad} (M)=0}$

Jacobson radical of rings

Let the following be a ring with one. ${\displaystyle R}$

definition

The Jacobson radical of the ring is defined as the Jacobson radical of the left module . It is noted as and characterized by the following equivalent conditions: ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle J(R)}$

• as the average of all maximum left / right ideals
• as the average of all cancellers of simple left - module / right- module ${\displaystyle R}$ ${\displaystyle R}$
• ${\displaystyle \{x\in R\mid \forall y\in R\colon 1-xy\in R^{\times }\}}$
• ${\displaystyle \{x\in R\mid \forall y,z\in R\colon 1-zxy\in R^{\times }\}}$
• ${\displaystyle \{x\in R\mid \forall z\in R\colon 1-zx\ \mathrm {ist\ linksinvertierbar} \}}$

properties

• The ring is semi- simple if and only if it is leftartinian . ${\displaystyle R}$ ${\displaystyle J(R)=0}$
• For any Linksartinian ring , the ring is semi-simple. ${\displaystyle R}$ ${\displaystyle R/J(R)}$
• Is linksartinsch, then for all -Linksmodul : . ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle J(R)M=\mathrm {Rad} (M)}$
• ${\displaystyle J(R)}$is the smallest ideal of having the property that is semi-simple. ${\displaystyle I}$ ${\displaystyle R}$ ${\displaystyle R/I}$
• Is a Nillinksideal by then: . ${\displaystyle N}$ ${\displaystyle R}$ ${\displaystyle N\subseteq J(R)}$
• If leftartinian, then it is a nilpotent ideal. ${\displaystyle R}$ ${\displaystyle J(R)}$
• If left-Artinian, then the Jacobson radical is the same as the prime radical . ${\displaystyle R}$
• With Zorn's lemma , the existence of maximal ideals follows for every ring , so for holds . ${\displaystyle R\neq \{0\}}$ ${\displaystyle R\neq \{0\}}$ ${\displaystyle J(R)\neq R}$

Examples

• The Jacobson radical of an oblique body is ; likewise the Jacobson radical of . ${\displaystyle \{0\}}$ ${\displaystyle \mathbb {Z} }$
• The Jacobson radical of is . ${\displaystyle \mathbb {Z} /24\mathbb {Z} }$ ${\displaystyle 6\mathbb {Z} /24\mathbb {Z} }$
• The Jacobson radical of the ring of all upper triangular matrices over a body contains those upper triangular matrices whose diagonal entries vanish. ${\displaystyle n\times n}$ ${\displaystyle K}$
• The Jacobson radical of every local ring is its maximum ideal, i.e. it consists precisely of its non-units.
• The Jacobson radical of a commutative Banach algebra is exactly the core of the Gelfand transformation .

literature

• KA Zhevlakov: Jacobson radical . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).