# properties of the Jacobson radical

Let $R,T$ be rings and $\varphi:R\rightarrow T$ be a surjective homomorphism. Then $\varphi(J(R))\subseteq J(T)$.

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 $1-t\varphi(a)$ is left invertible:

Since $\varphi$ is surjective, $t=\varphi(r)$ 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

 $\varphi(u)\left(\varphi(1)-\varphi(r)\varphi(a)\right)=\varphi(u)\varphi(1-ra)% =\varphi(1)=1$

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

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

Proof:
Let $\pi_{1}:R\times T\rightarrow R$ be a (surjective) projection. By the previous theorem, $\pi_{1}(J(R\times T))\subseteq J(R)$.

Similarly let $\pi_{2}:R\times T\rightarrow T$ 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 PropertiesOfTheJacobsonRadical 2013-03-22 12:49:43 2013-03-22 12:49:43 yark (2760) yark (2760) 12 yark (2760) Result msc 16N20