Rings of Invariant Module Type and Automorphism-Invariant Modules

Abstract

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 quasiinjective. 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 finitedimensional algebra over a field F 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 Rmodule is automorphism-invariant. In this paper we completely characterize an indecomposable right artinian ring of RAI-type.2000 Mathematics Subject Classification. 16U60, 16D50.

Additive Unit Representations in Endomorphism Rings and an Extension of a Result of Dickson and Fuller

“…Examples of automorphism-invariant modules which are not quasi-injective, can be found in [5] and [13]. And recently, it has been shown in [5] that a module M is automorphism-invariant if and only if every monomorphism from a submodule of M extends to an endomorphism of M. For more details on automorphism-invariant modules, see [5], [9], [11], and [12].…”

Injective and automorphism-invariant non-singular modules

“…In [3,Theorem 16], Er, Singh and Srivastava proved that a module M is automorphism-invariant if and only if M is a pseudo-injective module, i.e., for any submodule X of M, every monomorphism X → M can be extended to an endomorphism of the module M. Pseudo-injective modules were studied in several papers; e.g., see [8], [15], [3]. Automorphism-invariant modules were studied in several papers; e.g., see [1], [3], [5], [11], [13], [16], [17], [18], [19], [20].…”

Rings with each right ideal automorphism-invariant

“…Recently, Guil Asensio, Keskin Tütüncü and Srivastava [13] have initiated the study of a more general theory of modules invariant under automorphisms of their covers and envelopes. See [1], [15], [16], [37] and [38] for more details on automorphism-invariant modules.…”

“…Rings all of whose right ideals are automorphism-invariant are called right arings [38]. Since every quasi-injective module is automorphism-invariant, the family of right a-rings includes right q-rings.…”