I'm looking for an example of ring epimorphism $\varphi:R\rightarrow S$ such that the natural homomorphism $\tilde\varphi:J(R)\rightarrow J(S)$ is not surjective, where $J(R)$ is a Jacobson radical.
2026-04-08 06:04:17.1775628257
Properties of Jacobson radical
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$$\varphi:Z\rightarrow Z_8; \quad \varphi(n)=\bar n $$
$J(Z)=0$. so $\varphi(J(Z))=0$ but $J(Z_8)= \langle 2 \rangle$