Automorphism-invariant Modules Research Articles

On automorphism-invariant modules

Let M and N be two modules. M is called automorphism N-invariant if for any essential submodule A of N, any essential monomorphism f : A → M can be extended to some g ∈ Hom (N, M). M is called automorphism-invariant if M is automorphism M-invariant. This notion is motivated by automorphism-invariant modules analog discussed in a recent paper by Lee and Zhou [Modules which are invariant under automorphisms of their injective hulls, J. Algebra Appl. 12(2) (2013), 1250159, 9 pp.]. Basic properties of mutually automorphism-invariant modules and automorphism-invariant modules are proved and their connections with pseudo-injective modules are addressed.

Characteristic submodules of injective modules over strongly prime rings

It is proved that each automorphism-invariant module over a right strongly prime ring is injective, if it is not singular.

Automorphism-invariant modules satisfy the exchange property

Warfield proved that every injective module has the exchange property. This was generalized by Fuchs who showed that quasi-injective modules satisfy the exchange property. We extend this further and prove that a module invariant under automorphisms of its injective hull satisfies the exchange property. We also show that automorphism-invariant modules are clean and that directly-finite automorphism-invariant modules satisfy the internal cancellation and hence the cancellation property.

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

It is proved, among other results, that a prime right nonsingular ring (in particular, a simple ring) R is right self-injective if RR 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.

Characteristic submodules of injective modules

We study modules which are characteristic submodules in their injective hulls. The author is supported by the Russian Foundation for Basic Research, project 08-01- 00693-a All rings below are assumed to be associative and with nonzero identity ele- ment; all modules are assumed to be unitary. A moduleM is said to be Noetherian if every properly ascending chain of submodules in M is finite. Expressions of the form 'A is a Noetherian ring' mean that AA and AA are Noetherian modules. A moduleM is said to be injective with respect to the moduleX or X-injective if for any submodule X1 in X, every homomorphismX1 → M may be extended to a homomorphismX → M. For a ring A, an A-module M is said to be injective if M is injective with respect to any A-module. A submodule M of the module E is said to be essential if M ∩ E1 6 0 for any nonzero submodule E1 in E. If E is an injective module and M is an essential submodule in E then E is called the injective hull of the module M. The injective hull is unique up to isomorphism. A submoduleM of the module E is called a characteristic submodule if f(M) ⊆ M for every automorphismf of the module E. In the papers (3), (8), and (10), a module M is said to be automorphism- invariant if M is a characteristic submodule of the injective hull of M. It is easy to verify that a module M is an automorphism-invariant if and only if M is a charac- teristic submodule of some injective module; see Remark 6 below. Remark 1. In (3, Theorem 16), it is proved that a module M is an automorphism-invariant module if and only if M is a pseudo-injective module, i.e., for any submodule X in M , every monomorphism X → M may be exten- ded to an endomorphism of the moduleM . For example, see (6), (11), (3) about pseudo-injective modules. Remark 2. It is clear that every injective module is an automorphism-invariant module. The converse is not true, since every simple Abelian group is a finite automorphism-invariant non-injective module over the ring of integers Z.

MODULES WHICH ARE INVARIANT UNDER AUTOMORPHISMS OF THEIR INJECTIVE HULLS

A module is defined to be an automorphism-invariant module if it is invariant under automorphisms of its injective hull. Quasi-injective modules and, more generally, pseudo-injective modules are all automorphism-invariant. Here we develop basic properties of these modules, and discuss when an automorphism-invariant module is quasi-injective or injective. Some known results on quasi-injective and pseudo-injective modules are extended.

Dual automorphism-invariant modules

A module M is called an automorphism-invariant module if every isomorphism between two essential submodules of M extends to an automorphism of M. This paper introduces the notion of dual of such modules. We call a module M to be a dual automorphism-invariant module if whenever K1 and K2 are small submodules of M, then any epimorphism η:M/K1→M/K2 with small kernel lifts to an endomorphism φ of M. In this paper we give various examples of dual automorphism-invariant module and study its properties. In particular, we study abelian groups and prove that dual automorphism-invariant abelian groups must be reduced. It is shown that over a right perfect ring R, a lifting right R-module M is dual automorphism-invariant if and only if M is quasi-projective.