# properties of the Jacobson radical

Theorem:

Let $R,T$ be rings and $\phi :R\to T$ be a surjective^{} homomorphism^{}. Then $\phi (J(R))\subseteq J(T)$.

Proof:

We shall use the characterization of the Jacobson radical^{} as the set of all $a\in R$ such that for all $r\in R$, $1-ra$ is left invertible.

Let $a\in J(R),t\in T$. We claim that $1-t\phi (a)$ is left invertible:

Since $\phi $ is surjective, $t=\phi (r)$ for some $r\in R$. Since $a\in J(R)$, we know $1-ra$ is left invertible, so there exists $u\in R$ such that $u(1-ra)=1$. Then we have

$$\phi (u)\left(\phi (1)-\phi (r)\phi (a)\right)=\phi (u)\phi (1-ra)=\phi (1)=1$$ |

So $\phi (a)\in J(T)$ as required.

Theorem:

Let $R,T$ be rings. Then $J(R\times T)\subseteq J(R)\times J(T)$.

Proof:

Let ${\pi}_{1}:R\times T\to R$ be a (surjective) projection^{}.
By the previous theorem, ${\pi}_{1}(J(R\times T))\subseteq J(R)$.

Similarly let ${\pi}_{2}:R\times T\to T$ be a (surjective) projection. We see that ${\pi}_{2}(J(R\times T))\subseteq J(T)$.

Now take $(a,b)\in J(R\times T)$. Note that $a={\pi}_{1}(a,b)\in J(R)$ and $b={\pi}_{2}(a,b)\in J(T)$. Hence $(a,b)\in J(R)\times J(T)$ as required.

Title | properties of the Jacobson radical |
---|---|

Canonical name | PropertiesOfTheJacobsonRadical |

Date of creation | 2013-03-22 12:49:43 |

Last modified on | 2013-03-22 12:49:43 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 12 |

Author | yark (2760) |

Entry type | Result |

Classification | msc 16N20 |