Rings of Invariant Module Type and Automorphism-Invariant Modules

Abstract

SURJEET SINGH AND ASHISH K. SRIVASTAVADedicated to T. Y. Lam on his 70 th BirthdayAbstract. A module is called automorphism-invariant if it is invariant underany automorphism of its injective hull. In [Algebras for which every indecom-posable 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 isa finite-dimensional algebra over a field F with more than two elements thenan 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 fieldwith two elements. Dickson and Fuller had further shown that if Ris a finite-dimensional algebra over a field Fwith more than two elements, then R is ofright invariant module type if and only if every indecomposable right R-moduleis automorphism-invariant. We extend the result of Dickson and Fuller to anyright artinian ring. A ring R is said to be of right automorphism-invarianttype (in short, RAI-type) if every finitely generated indecomposable right R-module is automorphism-invariant. In this paper we completely characterizean indecomposable right artinian ring of RAI-type.