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).
2026-03-26 17:30:33.1774546233
PCI-ring which is not semisimple .
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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.
No: a commutative $V$- ring is Von Neumann regular, and a commutative VNR domain is a field.