Subrings of I-rings and S-rings (Q1380770)

In this paper, a left \(R\)-module \(M\) over a ring \(R\) is said to satisfy property I (property S) if every injective (resply. surjective) endomorphism of \(M\) is an automorphism. It is well-known that Artinian modules satisfy I and Noetherian modules satisfy S, but not conversely in general in either case. The ring \(R\) is called a left I-ring (left S-ring) if every left \(R\)-module with property I is Artinian (resply, with S is Noetherian). \textit{A. M. Kaidi} and the author proved earlier [Lect. Notes Math. 1328, 245-254 (1988; Zbl 0655.13013)]\ that a commutative ring \(R\) is an I-ring iff \(R\) is an Artinian principal ideal ring. In the present paper, the main result is that if \(R\) is a left I-ring (S-ring) and \(B\) is a subring of \(R\) contained in the centre of \(R\) then \(B\) is an I-ring (resply. S-ring) if \(R\) is a finitely generated flat \(B\)-module. This is proved by first showing that if \(R\) is a ring with a polynomial identity which is either a left I-ring or a left S-ring then \(R\) is left Artinian.