Kasch ring

In ring theory, a subfield of abstract algebra, a right Kasch ring is a ring "R" for which every simple right "R" module is isomorphic to a right ideal of "R". Analogously the notion of a left Kasch ring is defined, and the two properties are independent of each other.
Kasch rings are named in honor of mathematician Friedrich Kasch. Kasch originally called Artinian rings whose proper ideals have nonzero annihilators "S-rings". The characterizations below show that Kasch rings generalize S-rings.
Definition.
Equivalent definitions will be introduced only for the right-hand version, with the understanding that the left-hand analogues are also true. The Kasch conditions have a few equivalent statements using the concept of annihilators, and this article uses the same notation appearing in the annihilator article.
In addition to the definition given in the introduction, the following properties are equivalent definitions for a ring "R" to be right Kasch. They appear in :
Examples.
The content below can be found in references such as , , .