On the automorphism-invariance of finitely generated ideals and formal matrix rings

Abstract

In this paper, we study rings having the property that every finitely generated right ideal is automorphism-invariant. Such rings are called right f a-rings. It is shown that a right f a-ring with finite Goldie dimension is a direct sum of a semisimple artinian ring and a basic semiperfect ring. Assume that R is a right f a-ring with finite Goldie dimension such that every minimal right ideal is a right annihilator, its right socle is essential in RR, R is also indecomposable (as a ring), not simple, and R has no trivial idempotents. Then R is QF. In this case, QF-rings are the same as q?, f q?, a?, f a-rings. We also obtain that a right module (X,Y, f, g) over a formal matrix ring (R M N S) with canonical isomorphisms f? and g? is automorphism-invariant if and only if X is an automorphism-invariant right R-module and Y is an automorphism-invariant right S-module.