Abstract

A module M R is semisimple in case M is a direct sum Ц i ∈ I M i of a family of simple submodules {M i | i ∈ I}. A ring R is semisimple in case R R is. We have seen that any simple Artinian ring is semisimple. Finite ring products of simple Artinian rings are also semisimple. The Wedderburn-Artin theorem implies the converse: Every semisimple ring is isomorphic to a finite product of full matrix rings of various degrees over various fields. These rings are also characterized by the Wedderburn-Artin theorem (8.8) as right Artinian rings with no nilpotent ideals. (The condition is right-left symmetric.)

Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call