Jacobson图之间的同构
Isomorphisms between Jacobson graphs
摘要：
设$R$是具有非零单位元的交换环，$\mathfrak{J}_R$是它的Jacobson图.证明了若$R$和$R'$是有限交换环，则$\mathfrak{J}_R\cong\mathfrak{J}_{R'}$当且仅当$|J(R)|=|J(R')|$和$R/J(R)\cong R'/J(R')$ .对于Jacobson图$\mathfrak{J}_R$，我们得到了$\mathfrak{J}_R$的所有自同构的群$\mathrm{Aut}(\mathfrak{J}_R)$的结构，并证明了在一定条件下两个半单环$R$和$R'$同构当且仅当$\mathrm{Aut}(\mathfrak{J}_R)\cong\mathrm{Aut}(\mathfrak{J}_{R'})$ .
英文摘要：
Let $R$ be a commutative ring with a non-zero identity and $\mathfrak{J}_R$ be its Jacobson graph. We show that if $R$ and $R'$ are finite commutative rings, then $\mathfrak{J}_R\cong\mathfrak{J}_{R'}$ if and only if $|J(R)|=|J(R')|$ and $R/J(R)\cong R'/J(R')$. Also, for a Jacobson graph $\mathfrak{J}_R$, we obtain the structure of group $\mathrm{Aut}(\mathfrak{J}_R)$ of all automorphisms of $\mathfrak{J}_R$ and prove that under some conditions two semi-simple rings $R$ and $R'$ are isomorphic if and only if $\mathrm{Aut}(\mathfrak{J}_R)\cong\mathrm{Aut}(\mathfrak{J}_{R'})$.
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