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 .