LIFTING MODULES OVER RIGHT PERFECT RINGS

CHANG CHAEHOON AND YOSUKE KURATOMI

We consider associative rings with identity and all modules considered are unitary right R-modules. An R-module M is said to be an extending module, if it satisfies the following property: For any submodule X of M, there exists a direct summand of M which contains X as an essential submodule, that is, for any submodule X of M, there exists a closure of X in M which is a direct summand of M. Dually, M is called a lifting module, if it satisfies the dual property: For any submodule X of M, there exists a direct summand of M which is a co-essential submodule of X, that is, for any submodule X of M, there exists a co-closure of X in M which is a direct summand of M (cf.[4]). Any submodule X of M has a closure in M, but not always has a co-closure in M. It seems that the difficulty of study of lifting modules is hidden here.

Okado [3] has studied the decomposition of extending modules over right noetherian rings and obtained the following:

Theorem R is a right noetherian if and only if every extending R-module is expressed as a direct sum of indecomposable (uniform) modules.

As a dual problem, we consider the following problem:

Problem Which ring R has the property that every lifting R-module has an indecomposable decomposition ?

Our purpose of this note is to study this problem and show the following:

Result 1 Every lifting module over right perfect rings is expressed as a direct sum of indecomposable (cyclic hollow) modules.

Result 2 Let R be a right perfect ring and let M be a lifting module.

(1) If M is generalized M-projective, then M has the exchange property.

(2) If M has the finite exchange property, then M has the exchange property.

Result 3 Any lifting module over right perfect rings has the exchange property.

1. Preliminaries

Let M be a module. A submodule S of M is said to be a small submodule of M (denoted by S ≪ M) if M ≠ K + S for any proper submodule K of M. Let N and K be submodules of M with K ⊆ N. K is said to be a co-essential submodule of N in M if N/K ≪ M/K and we write K ⊆_{c} N in M in this case. Let X be a submodule of M. X is called co-closed submodule in M if X has no proper co-essential submodule in M. X′ is called a co-closure of X in M if X′ is a co-closed submodule of M with X′ ⊆_{c} X in M. K <_{⊕} N means that K is a direct summand of N.

The detailed version of this paper has been submitted for publication elsewhere.

A module M is called a lifting module if, for any submodule X of M, there exists a direct summand X^{∗} of M such that X^{∗} ⊆_{c} X.

Let {M_{i} | i ∈ I} be a family of modules and let M = ⊕_{I}M_{i}. M is said to be a lifting module for the decomposition M = ⊕_{I}M_{i} if, for any submodule X of M, there exist X^{∗} ⊆ M and \overline{M_{i}} ⊆ M_{i} (i ∈ I) such that X^{∗} ⊆_{c} X and M = X^{∗} ⊕ (⊕_{I} \overline{M_{i}}), that is, M is a lifting module and satisfies the internal exchange property in the direct sum M = ⊕_{I}M_{i}.

Let X be a submodule of M. A submodule Y of M is called a supplement of X in M if M = X + Y and X ∩ Y ≪ Y. Note that any supplement submodule (hence any direct summand) of a module M is co-closed in M.

A module M is called supplemented if, X contains a supplement of Y in M whenever M = X +Y. Note that the module M is lifting if and only if M is supplemented and every supplement submodule of M is a direct summand. An epimorphism P →^{f} M → 0 with P projective, is called a projective cover of M if ker f ≪ P. The notion of a projective cover is dual to that of an injective hull. A ring R is right perfect if every right R-module has a projective cover. Now we consider the following condition:

(∗) Any submodule of M has a co-closure in M. By [4, Theorem 1.3], we note that every module M over right perfect ring satisfies the condition (∗).

Proposition 1.1. A ring R is right perfect if and only if every projective right R-module is lifting.

Proof. Let P be a projective R-module. For any submodule A of P, consider a canonical epimorphism φ : P → P/A. Since P/A has a projective cover, there exists a decomposition P = P_{1} ⊕ P_{2} such that P_{2} ⊆ ker(φ) = A and (φ|P_{1}) : P_{1} → P/A → 0 a projective cover. Thus ker(φ|P_{1}) = P_{1} ⋂ A ≪ P, hence P is lifting. Conversely, since any module M is an epimorphic image of a free module, there exist a projective module P and epimorphism f : P → M. Since P is lifting, there exists a decomposition P = A^{∗} ⊕ A^{∗∗} such that A^{∗} ⊆_{c} ker(f) in P. Thus (f|_{A^{∗∗}}) : A^{∗∗} → M → 0 is a projective cover.

The following lemmas are due to Oshiro [4].

Lemma 1.2. Any projective module satisfies the following condition:

(D) If M_{1} and M_{2} are direct summands of M such that M_{1} ∩ M_{2} ≪ M and M = M_{1} + M_{2}, then M = M_{1} ⊕ M_{2}.

Lemma 1.3. If M is a lifting module with the condition (D), then M satisfies the following condition:

(D′) If M_{i} <_{⊕} M (i ∈ I), M = ∑_{I} M_{i} and M_{j} ∩ ∑_{i≠j} M_{i} ≪ M, then M = ∑_{I} M_{i} is direct.

2. Local Summands

∑ ⊕_{λ∈Λ}X_{λ} ⊆ X is called a local summand of X, if ∑ ⊕_{λ∈F} X_{λ} <_{⊕} X for every finite subset F ⊆ Λ.

The following lemma due to Oshiro is useful. For Okado’s result above, this lemma was used.

Lemma 2.1 (cf. [5]). If every local summand of M is a direct summand, then M has an indecomposable decomposition.

By Lemma 1.2 and [4, Proposition 3.2], the following holds:

Lemma 2.2 (cf. [4]). Every local summand of projective lifting modules is a direct summand.

A module M is said to have the (finite) exchange property if, for any (finite) index set I, whenever M ⊕ N = ⊕_{I}A_{i} for modules N and A_{i}, then M ⊕ N = M ⊕ (⊕_{I}B_{i}) for submodules B_{i} ⊆ A_{i}.

A module M has the (finite) internal exchange property if, for any (finite) direct sum decomposition M = ⊕_{I}M_{i} and any direct summand X of M, there exist submodules \overline{M_{i}} ⊆ M_{i} such that M = X ⊕ (⊕_{I} \overline{M_{i}}).

The following is essentially due to Harada [1].

Theorem 2.3. Let M = ⊕_{I}M_{α}, where each M_{α} has a local endomorphism ring. Then the following conditions are equivalent:

(1) M has the internal exchange property (in the direct sum M = ⊕_{I}M_{α} );

(2) M has the (finite) exchange property;

(3) Every local summand of M is a direct summand.

3. Main Results

The following is a main result of our talk.

Theorem 3.1. If R is a right perfect ring, then every local summand of lifting modules is direct summand.

Sketch of Proof: Let M be a lifting module and let ⊕_{I}X_{i} be a local summand of M. Since R is a right perfect ring, M has a projective cover, say ker f ≪ P →^{f} M → 0. By Lemma 1.1, P is projective lifting. So there exists a decomposition P = P_{i} ⊕P_{i}^{∗} such that P_{i} ⊆_{c} f^{−1}(X_{i}) in P. As X_{i} is co-closed in M, f(P_{i}) = X_{i}. By Lemma 3.1, ∑_{I} P_{i} = ⊕_{I}P_{i}. By Lemma 2.3, ⊕_{I}P_{i} <_{⊕} P. Since f(⊕_{I}P_{i}) is co-closed in M, we see

⊕_{I}X_{i} = f(⊕_{I}P_{i}) <_{⊕} M.

Thus any local summand of M is a direct summand.

□

The following result is a consequence of Lemma 2.1 and Theorem 3.1.

Theorem 3.2. Any lifting module over right perfect rings has an indecomposable decomposition.

A module H is said to be hollow if H is indecomposable lifting. By a proof of [9, Proposition 1], we see

Lemma 3.3. Let X be a hollow module. If X ⊕ X has the internal exchange property, then X has a local endomorphism ring.

By Theorem 3.2 and Lemma 3.3, the following holds:

Corollary 3.4. Let R be a right perfect ring and M be a lifting module. If M has the finite exchange property, then it has exchange property.

A module A is said to be generalized B-projective (B-cojective) if, for any homomorphism f : A → X and any epimorphism g : B → X, there exist decompositions A = A_{1} ⊕ A_{2}, B = B_{1} ⊕ B_{2}, a homomorphism h_{1} : A_{1} → B_{1} and an epimorphism h_{2} : B_{2} → A_{2} such that g ∘ h_{1} = f|_{A_{1}} and f ∘ h_{2} = g|_{B_{2}} (cf. [8]). The concept of generalized projective is a dual one of generalized injective (cf. [6]). Note that every B-projective modules is generalized B-projective. Lemma 3.5 (cf. [2]). Let M_{1} and M_{2} be lifting modules and put M = M_{1} ⊕ M_{2}. Assume that M satisfies the condition (∗). Then M is lifting for M = M_{1} ⊕ M_{2} if and only if M_{i} is generalized M_{j}-projective (i ≠ j).

By Lemma 3.5, 3.3, Theorem 3.1 and 2.3, we obtain the following: Corollary 3.6. Let R be a right perfect ring and M be a lifting module. If M is generalized M-projective, then M has the exchange property. In particular, any projective module over right perfect rings has the exchange property.

Remark Recently, we obtained that any hollow module over right perfect rings has a local endomorphism ring. Thus, by Theorem 2.3, the following holds: Any lifting module over right perfect rings has the exchange property.

References

[1] M. Harada: Factor categories with applications to direct decomposition of modules, LN Pure Appl. Math. 88, Dekker, New York (1983)

[2] Y. Kuratomi: Direct sums of lifting modules and internal exchange property, preprint.

[3] M. Okado: On the decomposition of extending modules, Math. Japonica 29 (1984), 939-941.

[4] K. Oshiro: Semiperfect modules and quasi-semiperfect modules, Osaka J. Math. 20 (1983), 337-372.

[5] K. Oshiro: Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. 13 (1984), 310-338.

[6] K. Hanada, Y. Kuratomi and K. Oshiro: On direct sums of extending modules and internal exchange property , Journal of Algebra 250 (2002), 115-133.

[7] S.H. Mohamed and B.J. Müller: Continuous and Discrete Modules, London Mathematical Society Lecture Note Series. 147

[8] S.H. Mohamed and B.J. Müller: Cojective modules, preprint.

[9] R.B. Warfield, Jr.: A Krull-Schmidt theorem for infinite sums of modules, Proc. Amer. Math. Soc. 22 (1969), 460-465.

Chang Chaehoon Department of Mathematical Sciences Yamaguchi University Yamaguchi 753-8512 JAPAN E-mail: b1216@yamaguchi-u.ac.jp

Yosuke Kuratomi Kitakyushu National College of Technology 5-20-1 Shii, Kokuraminami, Kitakyushu, Fukuoka, 802-0985 JAPAN E-mail: kuratomi@kct.ac.jp