I'm trying to understand links between Artinian property of a ring $R$ and its spectrum. I suppose that $R$ is noetherian.
This answer is perfect, particulary: TFAE
5) $R$ is Artinian
6) $\operatorname{Spec}(R)$ is discrete
7) $\operatorname{Spec}(R)$ is finite and discrete
It is clear for me that (5) is equivalent with $\operatorname{Spec}(R)$ is $T_1$ i.e. points are closed so I get: (6) $\Rightarrow$ (5)
It is clear that (7) $\Rightarrow$ (6).
Any idea to conclude?
Best Answer
Let $R$ be artinian, with Jacobson radical $J$. Then $J$ is nilpotent and $R/J$ is product of fields. We can lift these idempotents to $R$, and hence $R$ is a finite product of local rings.
Assume therefore that $R$ is local artinian, with maximal ideal $J$. Since $J$ is nilpotent, it is contained in every prime ideal, so is the unique prime. Thus $\mathrm{Spec}\,R$ is a single point.