Rings of Invariant Module Type and Automorphism-Invariant Modules

A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. In [Algebras for which every indecomposable right module is invariant in its injective envelope, Pacific J. Math., vol. 31, no. 3 (1969), 655-658] Dickson and Fuller had shown that if R is a finite-dimensional algebra over a field F with more than two elements then an indecomposable automorphism-invariant right R-module must be quasi-injective. In this paper we show that this result fails to hold if F is a field with two elements. Dickson and Fuller had further shown that if R is a finite-dimensional algebra over a field IF with more than two elements, then R is of right invariant module type if and only if every indecomposable right R-module is automorphism-invariant. We extend the result of Dickson and Fuller to any right artinian ring. A ring R is said to be of right automorphism-invariant type (in short, RAI-type) if every finitely generated indecomposable right R-module is automorphism-invariant. In this paper we completely characterize an indecomposable right artinian ring of RAI-type.