PI-RINGS WITH ARTINIAN PROPER CYCLICS ARE NOETHERIAN

Year 2013, Volume: 13 Issue: 13, 40 - 42, 01.06.2013

Adel N. Alahmadi

Abstract

Non-Artinian algebras over which proper cyclic right modules are
Artinian must be right Ore domains. It is shown that if R is a PI-ring whose
proper cyclic right R-modules are Artinian, then R is right Noetherian. In
particular, if G is a solvable group and each proper cyclic right K[G]-module
is Artinian, then the group algebra K[G] is Noetherian. It is also shown that
for a group algebra K[G], if every proper cyclic right K[G]-module is Artinian
and K-finite dimensional, then K[G] is Noetherian.

Keywords

group algebra, proper cyclic, Artinian ring, Noetherian ring