A NOTE ON AUTOMORPHISM LIFTABLE MODULES

A module M is said to be an automorphism liftable module if for each submodule N of M , every automorphism of the quotient M/N can be lifted to an endomorphism of M . In this work, some properties of automorphism liftable modules are investigated. Also, characterization for some special rings such as perfect, semiperfect and uniserial are given by using automorphism liftable modules. Mathematics Subject Classification (2010): 16D40, 16W20, 20K10

In fact, the lifting φ is an isomorphism [13]. Every semisimple module is a dual automorphism-invariant module. Quasi projective and pseudo projective modules are dual automorphism-invariant modules. Also, the Prüfer group Z(p ∞ ) is not a dual automorphism invariant module over the ring of integers [13].
In [14], Tuganbaev introduced a new notion namely automorphism-extendable module. A right module M is said to be an automorphism-extendable module if for each submodule N of M , every automorphism of the module N can be extended to an endomorphism of M . In [14], he proved that every automorphism-invariant module is an automorphism-extendable module. But the converse need not be true. For example, Z is an automorphism-extendable Z-module but it is not an automorphism-invariant Z-module [14].
The dual notion of automorphism-extendable module was introduced by Selvaraj and Santhakumar in [12]. Such modules were called as automorphism liftable modules. Also, they characterized some rings by the properties of automorphism liftable modules and they studied automorphism liftable modules with the summand sum and summand intersection properties. At the same time Abyzov and Truong [2] introduced the same concept in the name of dual automorphism-extendable module. In this paper, we discuss the question "When a direct sum of automorphism liftable modules will become automorphism liftable?". Also we introduce automorphism liftable cover to characterize some rings.

Automorphism liftable modules
Selvaraj and Santhakumar introduced the notion automorphism liftable module in [12], and discussed some basic properties of the same. Here we discuss some more basic properties of automorphism liftable modules.
The definition of automorphism liftable module as defined in [12] is (1) It is very clear that every quasi (pseudo) projective module is automorphism liftable.
(2) Let R = F 2 F 2 0 F 2 be a ring and M = Then M/N isomorphic to F 2 . Hence the only automorphism of M/N is the identity. Therefore M is an automorphism liftable module over the ring R.
(4) Over a perfect ring every flat module is an automorphism liftable module [12].
We know that if any two modules M and N with projective covers are isomorphic, then their projective covers are isomorphic. The converse part need not be true for all modules. However, here we try to solve the converse with certain condition by using automorphism liftable modules. Proof. Since M 1 and M 2 have projective covers, M 1 ⊕ M 2 has a projective cover Let ρ 1 : P 1 → P 2 be an automorphism and ρ 2 : which is an automorphism of and J 2 = φ 2 (J(P 2 )). Since J(P 1 ) and J(P 2 ) are small sumodules of P 1 and P 2 respectively, clearly, J 1 and J 2 are small submodules of M 1 and M 2 respectively.
The direct sum of two automorphism liftable modules need not be automorphism liftable. For example [12], Z 2 and Z 4 are automorphism liftable Z-modules. But, is not an automorphism liftable Z-module. This motivates us to rise the question "When a direct sum of automorphism liftable modules will become automorphism liftable?". Through following results we partially answer the above question.  From [15,Proposition 2.6], we have that any module has a quasi projective cover whenever, it has a projective cover. Then the following corollary follows from the above theorem.
Corollary 3.6. If a module M has a projective cover, then it has an automorphism liftable cover.
In general, the converse of above corollary need not be true. With some certain condition as given in Theorem 3.7, it will be true.

S. SANTHAKUMAR
By the projectivity of P , there exists a homomorphism λ : P → L 1 such that φ • λ = λ. Since P ⊕ L 1 is automorphism liftable module, P and L 1 are relatively projective to each other. Therefore the map λ : P → L 1 splits, i.e., L 1 is a direct summand of P . Hence φ : L 1 → M is a projective cover of M .
Recall that a ring R is said to be a (semi) perfect ring if every (finitely generated) module has a projective cover. In [5], Golan proved that a ring is perfect if and only if every module has a quasi projective cover. In [11], Selvaraj and Santhakumar characterized perfect ring by using dual automorphism invariant cover with certain condition. Also in [12], they proved that a ring is perfect if and only if every flat module is automorphism liftable. Here we characterized the same by using automorphism liftable cover.

Corollary 3.8. A ring R is (semi) perfect if and only if every (finitely generated)
module has an automorphism liftable cover.
Proof. Suppose every (finitely generated) module has an automorphism liftable cover. Since every (finitely generated) module can be written as an epimorphic image of a (finitely generated) projective module, by Theorem 3.7, R is a (semi) perfect ring.
Conversely, suppose R is a (semi) perfect ring. Then by Corollary 3.6, every (finitely generated) module M has an automorphism liftable cover.
In [6], Golan prove that a ring R is semiperfect if and only if for all n ≥ 1, every cyclic R n -module has a quasi projective cover if and only if there exists an n > 1 such that every cyclic R n -module has a quasi projective cover. Inspired by this, we characterize the semiperfect ring by automorphism liftable covers.
Theorem 3.9. For any ring R, the following are equivalent: (1) R is semiperfect; (2) For all n ≥ 1, every cyclic R n -module has an automorphism liftable cover; (3) There exists an n > 1 such that every cyclic R n -module has an automorphism liftable cover. Proof.
(1) ⇒ (2) It follows from the fact that if R is semiperfect so is R n for all n ≥ 1, by [7,Theorem 3].
(3) ⇒ (1) Let n > 1 be satisfy the condition that every cyclic R n -module has an automorphism liftable cover. Let S be a right ideal of R, S n the right ideal of R n consisting of all matrices with entries from S. Let e ij ∈ R n be the matrix with P ⊕ M , where M = e 11 R n /e 11 S n and P = n i=2 e ii R n . P is clearly R n -projective and the map λ : P → M which sends [a ij ] to e 21 [a ij ]+e 11 S n is an R n -epimorphism.
Since P ⊕ M has an automorphism liftable cover over R n , by Theorem 3.7, M has a projective cover φ : Q → M over R n . Then φ(Qe 11 ) = (φQ)e 11 = M e 11 which is isomorphic, as an R-module, to R/S. Therefore Q is R n -projective and so Qe 11 is R-projective [7]. The induced R-homomorphism φ : Qe 11 → R/S is then a projective cover, proving (1).  In [3], Byrd proved that a ring R is uniserial if and only if every quasi projective module is quasi injective; equivalently if every quasi-injective module is quasiprojective. Here we characterize uniserial rings by using automorphism liftable and automorphism extendable modules.
Theorem 3.11. For any ring R, the following are equivalent: (1) R is uniserial; (2) Every quasi projective module is automorphism extendable; (3) Every quasi injective module is automorphism liftable; (4) R is quasi-Frobenius and every finitely generated quasi projective module is automorphism extendable; (5) R is quasi-Frobenius and every finitely generated quasi injective module is automorphism liftable.