Structure Theorem for Non-commutative Artin rings The following structure theorem is well known: 

A commutative Artinian ring is a finite direct product of local Artinian rings. 

Do we have such/similar structure theorem for noncommutative artinian rings?
 A: I think the best analogy you can get is:

A right artinian ring is a finite direct product of indecomposable right Artinian rings.

Basically, one arrives at the result you mentioned in the OP by noting that you cannot have an infinite chain of pairwise orthogonal idempotents in a commutative Artinian ring. 
Every finite set of central orthogonal idempotents of $R$ which add up to 1 corresponds to a decomposition of the ring into factor rings $e_iRe_i$.
If a central idempotent $e$ cannot be written as the sum of two central orthogonal idempotents, then $eRe$ is indecomposable (in the sense of ring decompositions). Let's call this idempotent $e$ irreducible, for the duration of our conversation.
In any ring, you can try to write $1=\sum e_i$ as a sum of central irreducible idempotents, but sometimes irreducible ones do not exist, and sometimes these sets can be infinite. Under pretty mild finiteness conditions you can guarantee the existence of a finite set of central irreducible idempotents. Even "Noetherian" works, of course.
So why does "local" show up in the commutative case? Lemma: a commutative Artinian ring whose only idempotents are 0 and 1 is local. (Hence a commutative Artinian indecomposable ring is local.)
