Rings and modules which are stable under automorphisms of their injective hulls

Abstract

It is proved, among other results, that a prime right nonsingular ring (in particular, a simple ring) R is right self-injective if R R is invariant under automorphisms of its injective hull. This answers two questions raised by Singh and Srivastava, and Clark and Huynh. An example is given to show that this conclusion no longer holds when prime ring is replaced by semiprime ring in the above assumption. Also shown is that automorphism-invariant modules are precisely pseudo-injective modules, answering a recent question of Lee and Zhou. Furthermore, rings whose cyclic modules are automorphism-invariant are investigated.

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

“…Thus, the result of Dickson and Fuller states that if R is a finite-dimensional 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. 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].…”

Rings with each right ideal automorphism-invariant

“…Example 2.2. Consider the ring R consisting of all eventually constant sequences of elements from F 2 (see [10,Example 9]). Clearly, R is a commutative automorphism-invariant ring as the only automorphism of its injective envelope is the identity automorphism.…”

“…Proof. By [10,Theorem 3], there exists a decomposition R R = A ⊕ B ⊕ C where A ∼ = B and the module C is square-free which is orthogonal to A ⊕ B. Let X := A ⊕ B and Y := C. Now we proceed to show that X is square-full.…”

“…These modules have been recently named as automorphism-invariant modules by Lee and Zhou in [31]. In [10] Er, Singh and Srivastava proved that a module M is automorphism-invariant if and only if any monomorphism from a submodule of M to M extends to an endomorphism of M thus showing that automorphism-invariant modules are precisely the pseudo-injective modules studied by Jain and Singh in [24] and Teply in [40]. Guil Asensio and Srivastava [14] have shown that automorphism-invariant modules satisfy the full exchange property and these modules also provide a new class of clean modules.…”

“…Guil Asensio and Srivastava [14] have shown that automorphism-invariant modules satisfy the full exchange property and these modules also provide a new class of clean modules. The decomposition of automorphism-invariant modules has been described in [10]. If M is an automorphism-invariant module, then M has a decomposition M = A ⊕ B where A is quasi-injective and B is square-free.…”