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.[1] Analogously the notion of a left Kasch ring is defined, and the two properties are independent of each other.
Kasch originally called Artinian rings whose proper ideals have nonzero annihilators S-rings.
[2][3] The characterizations below show that Kasch rings generalize S-rings.
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.