Semisimple Rings
A ring is said to be (left)-semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary, and one can speak of semisimple rings without ambiguity.
Semisimple rings are of particular interest to algebraists. For example, if the base ring R is semisimple, then all R-modules would automatically be semisimple. Furthermore, every simple (left) R-module is isomorphic to a minimal left ideal of R, that is, R is a left Kasch ring.
Semisimple rings are both Artinian and Noetherian. From the above properties, a ring is semisimple if and only if it is Artinian and its radical is zero.
If an Artinian semisimple ring contains a field, it is called a semisimple algebra.
Read more about this topic: Semisimple Module
Famous quotes containing the word rings:
“Ye say they all have passed away,
That noble race and brave;
That their light canoes have vanished
From off the crested wave;
That, mid the forests where they roamed,
There rings no hunters’ shout;
But their name is on your waters,
Ye may not wash it out.”
—Lydia Huntley Sigourney (1791–1865)