0
$\begingroup$

Artin-Wedderburn theorem states that a ring is semisimple iff it is a finite direct product of matrix rings of division rings. Rotman wrote in his text, Advanced modern algebra, that Wedderburn first proved the above statement for semisimple $k$-algebras where $k$ is a field, then later Artin generalized this as it is stated here, and this is why artinian rings are so called.

I don't get this last sentence. Of course, this might be just the author's opinion so there is no complete answer for this, but the problem is, I don't have any clue why this theorem makes it reasonable to call rings with DCC as artinian rings. What would be a reason?

$\endgroup$
1
$\begingroup$

The sentence just means that they are named after Artin, because he made important contributions to their theory. Note that every semisimple ring is necessarily Artinian.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.