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}$${\ displaystyle \ mathrm {Rad} (M) = \ {x \ in M ​​| x \ \ mathrm {is \ {\ ddot {u}} overfl {\ ddot {u}} ssig \ in} \ M \}}$${\ 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.${\ displaystyle M}$ ${\ displaystyle \ mathrm {Rad} (M) = 0}$

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}$
• ${\ displaystyle \ {x \ in R \ mid \ forall y \ in R \ colon 1-xy \ in R ^ {\ times} \}}$
• ${\ displaystyle \ {x \ in R \ mid \ forall y, z \ in R \ colon 1-zxy \ in R ^ {\ times} \}}$
• ${\ displaystyle \ {x \ in R \ mid \ forall z \ in R \ colon 1-zx \ \ mathrm {is \ left-invertible} \}}$

### 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 .