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

Abstract

Abstract. A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. Dickson and Fuller have 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 note, we extend and simplify the proof of this result by showing that any automorphism-invariant module over an algebra over a field with more than two elements is quasi-injective. Our proof is based on the study of the additive unit structure of endomorphism rings.

Automorphism-Invariant Modules

“…In [16] Guil Asensio and Srivastava extended the above mentioned result of Faith and Utumi to automorphism-invariant modules by proving that Proposition 10. [16] Let M be an automorphism-invariant module and R = End(M ). Then ∆ = {f ∈ R : Ker(f ) ⊆ e M } is the Jacobson radical of R, R/J(R) is a von Neumann regular ring and idempotents lift modulo J(R).…”

“…In [16] Guil Asensio and Srivastava extended the above mentioned result of Faith and Utumi to automorphism-invariant modules by proving that Proposition 10. [16] Let M be an automorphism-invariant module and R = End(M ).…”

Additive unit structure of endomorphism rings and invariance of modules

“…Therefore, End(M ) cannot have a homomorphic image isomorphic to F 2 . This means that M must be quasi-injective by Theorem 3.4 (see also [14,Theorem 3]), thus contradicting the assertion in [18, p. 362] that M is not quasi-injective.…”

“…It is then natural to ask when an automorphism-invariant module is quasiinjective. It has been shown in [14] that if M is a right R-module such that End R (M ) has no homomorphic image isomorphic to F 2 , then M is quasi-injective if and only it is automorphism-invariant. In particular, this is the case when M is a module over an Falgebra, where F is a field with more than two elements; thus extending the previously mentioned result of Dickson and Fuller for indecomposable modules [5].…”

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.…”