Is there an example of finite right(left) PCI-ring which is not right(left) semisimple ring? (A ring R is called right(left) PCI-ring if each proper right(left) cyclic R-module C is injective, proper cyclic means that C is cyclic but C is not isomorphic to R).
PCI-ring which is not semisimple .
2
$\begingroup$
abstract-algebra
ring-theory
modules
noncommutative-algebra
injective-module
1 Answers
2
By the Faith-Cozzens theorem, if it is not semisimple, then it is simple (and right Hereditary and right Ore and a $V$-domain.) But a finite and simple ring is semisimple. So, no, there is no such finite ring.
Thanks for answering. May I know if there exist a commutative PCI-ring not semisimple ring.
No: a commutative $V$- ring is Von Neumann regular, and a commutative VNR domain is a field.
-
0Thanks for answering. May I know if there exist a commutative PCI-ring not semisimple ring. – 2017-02-27
-
0@MManhal added this to the answer. If I remember right, the examples of nonsemisimple PCI rings are fairly rare. – 2017-02-28