# a ring modulo its Jacobson radical is semiprimitive

Let $R$ be a ring. Then $J(R/J(R))=(0)$.

Proof:

We will only prove this in the case where $R$ is a unital ring
(although it is true without this assumption^{}).

Let $[u]\in J(R/J(R))$.
By one of the characterizations^{} of the Jacobson radical^{},
$1-[r][u]$ is left invertible for all $r\in R$,
so there exists $v\in R$ such that $[v](1-[r][u])=1$.

Then $v(1-ru)=1-a$ for some $a\in J(R)$. There is a $w\in R$ such that $w(1-a)=1$, and we have $wv(1-ru)=1$.

Since this holds for all $r\in R$, it follows that $u\in J(R)$, and therefore $[u]=0$.

Title | a ring modulo its Jacobson radical is semiprimitive |
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Canonical name | ARingModuloItsJacobsonRadicalIsSemiprimitive |

Date of creation | 2013-03-22 12:49:34 |

Last modified on | 2013-03-22 12:49:34 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 13 |

Author | yark (2760) |

Entry type | Theorem^{} |

Classification | msc 16N20 |