In algebra , an artinian ring R is a zero dimensional noetherian ring.
With this definition the spectrum X of R equal to maximal spectrum equal to the set of minimal prime ideals which is finite, so the Zariski topology is the discrete topology.
If we combine this remark with the fact that idempotent of R are in bijection with clopen (open and closed) set of X, we get a proof that Artinian ring is product of local Artinian ring.
I'm interested in some criterion exemple when a ring a product of local ring ,so i try to express this result in language of scheme.
Precisely in artinian affine scheme(and i think more generally when the topology is finite discrete), we can glue germs over closed point to get a global section.
Do you know another proof of this result or some other condition for this to be true .
This look to me like some sheaf or homological problem
And thanks.
Best Answer
A commutative ring with identity is a product of local rings iff it is a semiperfect ring.
Since semiperfectness is related to projective coverings for f.g. modules, maybe this fits in with your suspicion that it's related to homological reasons.