Indian J. pure appl. Math., 38(3): 129-142, June 2007 °c Printed in India.

RINGS CLOSE TO BAER

LIXIN MAO Institute of Mathematics, Nanjing Institute of Technology, Nanjing 211167, Peoples’ Republic of China Department of Mathematics, Nanjing University, Nanjing 210093, Peoples’ Republic of China e-mail: [email protected]

(Received 19 January 2006; after final revision 25 September 2006; accepted 30 October 2006)

In this paper, we define two kinds of dimensions, called A-injective dimension and A- flat dimension, which measure how far away a ring is from being a Baer ring. These dimensions have nice properties when the ring in question is an AC ring, where R is said to be a left AC ring in case the left annihilator of each nonempty subset of R is a cyclic left ideal.

Key Words: A-Injective Dimension; A-Flat Dimension; AC Ring; A-; A-Flat Module; Preenvelope; Precover

1. NOTATION

In this section, we recall some known notions and facts needed in the sequel. Let C be a class of R-modules and M an R-module. Following [8], we say that a homomorphism 0 φ : M → C is a C-preenvelope if C ∈ C and the abelian group homomorphism HomR(φ, C ): 0 0 0 HomR(C,C ) → HomR(M,C ) is an epimorphism for each C ∈ C.A C-preenvelope φ : M → C is said to be a C-envelope if every endomorphism g : C → C such that gφ = φ is an isomorphism. Dually we have the definitions of a C-precover and a C-cover. C-envelopes (C-covers) may not exist in general, but if they exist, they are unique up to isomorphism. Specializing C to the class of injective modules and projective modules respectively, C-envelopes and C-covers agree with the usual injective envelopes and projective covers respectively (see [27])).

⊥ 1 Given a class L of R-modules, we will denote by L = {C : ExtR(L, C) = 0 for all L ∈ L} ⊥ 1 the right orthogonal class of L, and by L = {C : ExtR(C,L) = 0 for all L ∈ L} the left orthogonal class of L. 130 LIXIN MAO

A pair (F, C) of classes of R-modules is called a cotorsion theory [9] if F ⊥ = C and ⊥C = F. A cotorsion theory (F, C) is called perfect [13] if every R-module has a C-envelope and an F- cover. A cotorsion theory (F, C) is called complete [25] if for any R-module M, there are exact sequences 0 → M → C → F → 0 with C ∈ C and F ∈ F, and 0 → K → G → M → 0 with G ∈ F and K ∈ C. Obviously, if (F, C) is a complete cotorsion theory, then every R-module has a C-preenvelope and an F-precover.

0 00 A cotorsion theory (F, C) is said to be hereditary [11, 13] if whenever 0 → L → L → L → 0 00 0 is exact with L, L ∈ F, then L is also in F. By [10, Proposition 1.2], a cotorsion theory (F, C) is 0 00 0 00 hereditary if and only if whenever 0 → C → C → C → 0 is exact with C,C ∈ C, then C is also in C.

Throughout this paper, R is an associative ring with identity and all modules are unitary. MR + (RM) denotes a right (left) R-module. For an R-module M, the character module M is defined + by M = HomZ(M, Q/Z), pd(M) and fd(M) stand for the projective and flat dimension of M. lD(R) (wD(R)) stands for the left (the weak) global dimension of R. By the terminology “left annihilator I”, we will mean that there exists a nonempty subset S of R such that I is a left annihilator of S in R. For a ∈ R, l(a) denotes the left annihilator of a in R.

n Let M and N be R-modules. Hom(M,N) (resp. Ext (M,N)) means HomR(M,N) (resp. n R ExtR(M,N)), and similarly M ⊗ N (resp. Torn(M,N)) denotes M ⊗R N (resp. Torn (M,N)) for an integer n ≥ 1.

For unexplained concepts and notations, we refer the reader to [1, 9, 18, 22, 26, 27].

2. INTRODUCTION

Recall that a ring R is called Baer if the left annihilator of each nonempty subset of R is generated, as a left ideal, by an of R. It is easy to see that the Baer property is left-right symmetric (see [17]). Examples of Baer rings include domains, right (or left) Noetherian right (or left) PP rings, and right (or left) self-injective von Neumann regular rings. The study of Baer rings has its roots in . Kaplansky introduced Baer rings to abstract various properties of von Neumann algebras and complete ∗-regular rings [17]. Baer rings have been investigated by several authors (e.g., [2, 4, 14, 17, 18, 19, 21, 23]).

The main goal of this paper is to define some kinds of homological dimensions to measure how far away a ring is from being a Baer ring. Let us describe the contents of the paper in more detail.

In Section 3, the concept of A-injective modules and A-flat modules are first introduced. A left R-module M is said to be A-injective if Ext1(R/I, M) = 0 for any left annihilator I. A right

R-module N is called A-flat if Tor1(N, R/I) = 0 for any left annihilator I. Several elementary properties of A-flat and A-injective modules are obtained.

In Section 4, we define the concept of AC rings, which is a generalization of Baer rings. RINGS CLOSE TO BAER 131

R is said to be a left AC ring in case the left annihilator of each nonempty subset of R is a cyclic left ideal. We obtain some special properties of A-injective modules and A-flat modules over AC rings. For example, it is proven that, if R is a left AC ring, then every right R-module has an AF-preenvelope, and every left R-module has an AI-cover, where AI stands for the class of all A-injective left R-modules, and AF denotes the class of all A-flat right modules.

In Section 5, we first introduce left A-injective dimension and right A-flat dimension for rings, denoted by l.ai−D(R) and r.af −D(R) respectively. The A-injective dimension measures how far away a ring is from being a Baer ring. Then we show that these dimensions have the properties that we expect of a “dimension” when the ring in question is a left AC ring. For example, it is true that l.ai − D(R) = r.af − D(R) for any left AC ring R. It is proven that the following are equivalent for a left AC ring R: (1) l.ai − D(R) ≤ 1. (2) Every left R-module has a monic AI-cover. (3) Every right R-module has an epic AF-envelope. We also show that the following are equivalent for a left AC ring R: (1) l.ai − D(R) ≤ 2. (2) Every left R-module has an AI-cover with the unique mapping property. Finally, we get that, if a left AC ring R satisfies one of the following equivalent conditions: (1) R is a left A-injective ring; (2) Every right R-module has a monic AF-preenvelope; (3) Every left R-module has an epic AI-cover, then l.ai − D(R) = 0 or ∞.

3. A-INJECTIVE MODULESAND A-FLAT MODULES

We start with the following

Definition 3.1 — Let R be a ring. A left R-module M is said to be A-injective if Ext1(R/I, M)

= 0 for any left annihilator I. A right R-module N is called A-flat if Tor1(N, R/I) = 0 for any left annihilator I.

Clearly, any flat (injective) module is A-flat (A-injective). However, the converse is not true in general. For example, any module over the ring Z of integers is A-flat (A-injective). But not all Z -modules are flat (injective).

The next lemma will be used frequently in the sequel.

Lemma 3.2 — The following are equivalent for a right R-module M:

(1) M is A-flat.

(2) M + is A-injective.

1 + ∼ + PROOF : It holds by the standard isomorphism: Ext (R/I, M ) = Tor1(M, R/I) for any left annihilator I. ¤

In what follows, AI stands for the class of all A-injective left R-modules, and AF denotes the class of all A-flat right modules.

Proposition 3.3 — The following are true for any ring R:

1. AI is closed under extensions, direct products, direct summands; 132 LIXIN MAO

2. AF is closed under extensions, direct sums and direct summands.

PROOF : It is straightforward. Theorem 3.4 — Let R be a ring. Then

1. (⊥AI, AI) is a complete cotorsion theory.

2. (AF, AF ⊥) is a perfect cotorsion theory.

PROOF : (1). Let X be the set of representatives of all left R-modules R/I with I any left annihi- lator. Then AI = X⊥, and so (⊥AI, AI) is a cotorsion theory cogenerated by the set X. Thus the result follows from [7, Theorem 10] and [9, Definition 7.1.5].

(2) follows from [25, Lemma 1.11 and Theorem 2.8]. ¤

4. AC RINGS

Definition 4.1 — R is said to be a left AC ring in case for any nonempty subset X of R, the left annihilator of X in R is a cyclic left ideal. Similarly, we have the concept of right AC rings.

Obviously, Baer rings are left and right AC rings. But the converse is not true in general. For example, Z4 is an AC ring which is not a Baer ring.

Recall that a left R-module M is called FP -injective [24] if Ext1(N,M) = 0 for all finitely presented left R-modules N. R is called a left (right) FP -injective ring if RR (RR) is FP -injective.

Proposition 4.2 — The following are equivalent for a ring R:

(1) R is a Baer and right FP -injective ring. (2) R is a Baer and left FP -injective ring. (3) R is a von Neumann regular and right AC ring. (4) R is a von Neumann regular and left AC ring.

PROOF : (1) ⇒ (3) and (4). Every cyclic left ideal I is a left annihilator of a nonempty subset of R by [15, Corollary 2.5] since R is right FP -injective, and so I is a direct summand of RR since R is a Baer ring. Thus R is a . The others are easy. ¤ Although the concept of Baer rings is left-right symmetric, the next example shows that it is not true for AC rings.

Example 4.3 — Let K be a field with a subfield L such that dimLK = ∞, and there exists a

field isomorphism ϕ : K → L (for instance, K = Q(x1, x2, x3, ···),L = Q(x2, x3, ···)). Let R = K × K with multiplication

(x, y)(x0, y0) = (xx0, ϕ(x)y0 + yx0) x, y, x0, y0 ∈ K. RINGS CLOSE TO BAER 133

Then it is easy to see that R has exactly three right ideals: 0, R and (0,K). Therefore R is a right AC ring. On the other hand, let a = (0, 1) ∈ R. Then l(a) is not finitely generated (see [18, Example 4.46 (e)]). Thus R is not a left AC ring.

Next we discuss special properties of left A-injective modules and right A-flat modules over a left AC ring.

Proposition 4.4 — Let R be a left AC ring. Then (⊥AI, AI) and (AF, AF ⊥) are hereditary cotorsion theories.

PROOF : Let 0 → A → B → C → 0 be an exact sequence of left R-modules with A and B A-injective. For any left annihilator I, I = Ra =∼ R/l(a) for some a ∈ R since R is a left AC ring. Thus Ext2(R/I, A) =∼ Ext1(I,A) =∼ Ext1(R/l(a),A) = 0 since A is A-injective. On the other hand, we have the exact sequence 0 = Ext1(R/I, B) → Ext1(R/I, C) → Ext2(R/I, A) since B is A-injective. Thus Ext1(R/I, C) = 0, and so C is A-injective. It follows that (⊥AI, AI) is a hereditary cotorsion theory. ¤ Let 0 → M → N → L → 0 be an exact sequence with N and L A-flat. Then we get an induced exact sequence 0 → L+ → N + → M + → 0. Note that N + and L+ are A-injective by Lemma 3.2, so M + is A-injective by the above proof, and hence M is A-flat. Therefore (AF, AF ⊥) is a hereditary cotorsion theory.

Theorem 4.5 — The following are true for a left AC ring R: (1) AI is closed under pure submodules. (2) AF is closed under pure submodules. (3) AF is closed under direct products. (4) AI is closed under direct limits. (5) A left R-module M is A-injective if and only if M + is A-flat. (6) A left R-module M is A-injective if and only if M ++ is A-injective. (7) A right R-module M is A-flat if and only if M ++ is A-flat.

PROOF : (1) Let N be a pure submodule of an A-injective left R-module M. For any left annihilator I, we have the exact sequence

Hom(R/I, M) → Hom(R/I, M/N) → Ext1(R/I, N) → 0.

But the sequence Hom(R/I, M) → Hom(R/I, M/N) → 0 is exact since R/I is finitely presented and N is a pure submodule of M. Therefore Ext1(R/I, N) = 0 and so N is A-injective. (2) Let A be a pure submodule of an A-flat right R-module B, then the pure exact sequence 0 → A → B → B/A → 0 induces the split exact sequence 0 → (B/A)+ → B+ → A+ → 0. Thus A+ is A-injective since B+ is A-injective by Lemma 3.2. So A is A-flat.

(3) Let (Mi)i∈I be a family of A-flat right R-modules and I a left annihilator. Then we have a commutative diagram with exact rows: 134 LIXIN MAO

Y Y Y 0 −→ Tor1( Mi, R/I) −→ ( Mi) ⊗ I −→ ( Mi) ⊗ R γ↓ α ↓ β ↓ Y Y Y 0 −→ Tor1(Mi, R/I) −→ (Mi ⊗ I) −→ (Mi ⊗ R).

Note that I is finitely presented since R is a left AC ring. Thus α is an isomorphism by [9, The- Q orem 3.2.22], and so γ is an isomorphism by Five Lemma. Hence Tor1( Mi, R/I) = 0 since Q Q Tor1(Mi, R/I) = 0. That is, Mi is A-flat.

(4) Let I be any left annihilator, and (Mi)i∈J a family of A-injective left R-modules, where J is a directed set. Then we have a commutative diagram with exact rows:

1 Hom(R, lim Mi) −→ Hon(I, lim Mi) −→ Ext (R/I, lim Mi) −→ 0 → → → α↓ β ↓ γ ↓ 1 lim Hom(R,Mi) −→ lim Hom(I,Mi) −→ lim Ext (R/I, Mi) −→ 0. → → → Note that I is finitely presented since R is a left AC ring. So β is an isomorphism by [16, Propo- 1 ∼ sition 2.5], and hence γ is an isomorphism by Five Lemma. It follows that Ext (R/I, lim Mi) = → 1 lim Ext (R/I, Mi) = 0. That is, lim Mi is A-injective. → → + ∼ 1 + (5) Note that Tor1(M , R/I) = Ext (R/I, M) for any left R-module M and any left annihi- lator I by [5, Lemma 2.7 (2)] since R is a left AC ring. So M is A-injective if and only if M + is A-flat.

(6) If M is an A-injective left R-module, then M + is A-flat by (5) and hence M ++ is A- injective by Lemma 3.2. Conversely, if M ++ is A-injective, then M is A-injective by (1) since M is a pure submodule of M ++.

(7) If M is an A-flat right R-module, then M + is A-injective and hence M ++ is A-flat by (5). Conversely, if M ++ is A-flat, then M is A-flat by (2) since M is a pure submodule of M ++. ¤

The next lemma is a special case of [3, Theorem 5].

Lemma 4.6 – Let R be an arbitrary ring. Then for each cardinal λ, there is a cardinal κ such that for any left R-module M and for any L ≤ M such that Card(M) ≥ κ and Card(M/L) ≤ λ, the submodule L contains a non-zero submodule that is pure in M.

Theorem 4.7 — The following are true for a left AC ring R:

(1) Every right R-module has an AF-preenvelope.

(2) Every left R-module has an AI-cover.

PROOF : (1) Let N be any right R-module. By [9, Lemma 5.3.12], there is a cardinal number

ℵα such that for any homomorphism g : N → L with L A-flat, there is a pure submodule Q of L RINGS CLOSE TO BAER 135

such that Card(Q) ≤ ℵα and g(N) ⊆ Q. Note that Q is A-flat by Theorem 4.5 (2), and so N has an AF-preenvelope by Theorem 4.5 (3) and [9, Proposition 6.2.1].

(2) The proof is motivated by that of [20, Lemma 4.8]. Suppose that N is a left R-module with Card(N) = λ. Let κ be a cardinal as in Lemma 4.6. By [9, Proposition 5.2.2], we only need to show that any homomorphism f : D → N with D ∈ AI has a factorization D → C → N with C ∈ AI and Card(C) ≤ κ.

Without loss of the generality, we may assume that Card(D) ≥ κ. Note that Card (D/K) ≤ λ since D/K embeds in N, where K = ker(f). Thus K contains a non-zero submodule D0 which is pure in D by Lemma 4.6. Note that D0 is A-injective by Theorem 4.5 (1) and so D/D0 is A-injective by Proposition 4.4.

If Card(D/D0) ≤ κ, then we have done since f factors through D/D0. If Card(D/D0) > κ, then there exists g : D/D0 → N with ker(g) = K1/D0. So K1/D0 contains a non-zero submodule ∼ D1/D0 which is pure in D/D0 by Lemma 4.6. Therefore D/D1 = (D/D0)/(D1/D0) is A- injective.

If Card(D/D1) ≤ κ, then we have done since f factors through D/D1. If Card(D/D1) > κ, then continuing the process above, we will arrive at lim(D/Di) such that Card(lim(D/Di)) ≤ κ. → → Note that lim(D/Di) is A-injective by Theorem 4.5 (4), and f factors through lim(D/Di). So N → → has an AI-precover, and hence has an AI-cover by [9, Corollary 5.2.7].

Proposition 4.8 — The following are equivalent for a left AC ring R: (1) Every A-flat right R-module is flat. (2) Every A-injective left R-module is FP -injective.

PROOF : (1) ⇒ (2). Let M be any A-injective left R-module. Then M + is A-flat by Theorem 4.5 (5), and so M + is flat by (1). On the other hand, by [5, Lemma 2.7 (1)], for any finitely presented left R-module N, there is an exact sequence

+ 1 + Tor1(M ,N) → (Ext (N,M)) → 0.

Thus Ext1(N,M) = 0, and so M is FP -injective.

(2) ⇒ (1). Let M be any A-flat right R-module. Then M + is A-injective by Lemma 3.2, and so M + is FP -injective by (2). Hence M is flat. ¤

5. A-INJECTIVE DIMENSIONAND A-FLAT DIMENSION

Definition 5.1 — Let M be a left R-module and aid(M) denote the smallest integer n ≥ 0 such that Extn+1(R/I, N) = 0 for any left annihilator I. We call aid(M) the A-injective dimension of M. If no such n exists, set aid(M) = ∞.

Put l.ai−D(R) = sup{aid(M): M is a left R-module} and call l.ai−D(R) the left A-injective dimension of the ring R. 136 LIXIN MAO

Let N be a right R-module and afd(N) stand for the smallest integer n ≥ 0 satisfying that

Torn+1(N, R/I) = 0 for any left annihilator I. We call afd(N) the A-flat dimension of N. If no such n exists, set afd(N) = ∞.

Put r.af − D(R) = sup{afd(M): M is a right R-module} and call r.af − D(R) the right A-flat dimension of the ring R.

Theorem 5.2 — The following are equivalent for any ring R: (1) l.ai − D(R) = 0. (2) R is a Baer ring. (3) Every left R-module is A-injective. (4) Every cyclic torsionless left R-module is projective.

PROOF : (1) ⇔ (3) and (2) ⇒ (3) are clear. (3) ⇒ (2). Let I be any left annihilator. Then Ext1(R/I, I) = 0 by (3), and so the exact sequence 0 → I → R → R/I → 0 is split. Thus I is a direct summand of RR. Hence R is a Baer ring.

(3) ⇔ (4) follows from the fact that a left ideal I is a left annihilator in R if and only if R/I is a torsionless left R-module.

Remark 5.3 : (1) By Theorem 5.2, l.ai − D(R) measures how far away a ring R is from being a Baer ring.

(2) Clearly, l.ai − D(R) ≤ lD(R) and r.af − D(R) ≤ wD(R).

The above equalities hold if every left ideal is a left annihilator.

The above inequalities may be strict. For example, l.ai − D(Z) = r.af − D(Z) = 0, but lD(Z) = wD(Z) = 1.

(3) It is easy to verify the following equalities:

l.ai − D(R) = sup{pd(M): M is a cyclic torsionless left R-module} and

r.af − D(R) = sup{fd(M): M is a cyclic torsionless left R-module}.

Thus r.af − D(R) ≤ l.ai − D(R) for any ring R.

Proposition 5.4 — Let R be a left AC ring. The following are equivalent for any left R-module M and an integer n ≥ 0:

(1) aid(M) ≤ n. (2) Extn+1(R/I, M) = 0 for any left annihilator I.

(3) Extn+j(R/I, M) = 0 for any left annihilator I and any j ≥ 1.

(4) There exists an exact sequence 0 → M → E0 → E1 → · · · → En → 0, where each Ei is A-injective. RINGS CLOSE TO BAER 137

PROOF : (1) ⇒ (2). Suppose aid(M) = m ≤ n. Let I be a left annihilator. Since R is a left ∼ ∼ AC ring, there exist left annihilators I1,I2, ··· ,In−m (write I0 = I) such that I = R/I1,I1 = ∼ n+1 ∼ n ∼ n ∼ R/I2, ··· ,In−m = R/In−m+1. Therefore Ext (R/I, M) = Ext (I,M) = Ext (R/I1,M) = n−1 ∼ ∼ m+1 Ext (I1,M) = ··· = Ext (R/In−m,M) = 0.

(2) ⇒ (3) is easy by induction.

(3) ⇒ (4). There exists an exact sequence 0 → M → E0 → E1 → · · · → En−1 → L → 0, where each Ei is injective. Thus L is A-injective by dimension shifting.

(4) ⇒ (1) is obvious by dimension shifting.

Proposition 5.5 — Let R be a left AC ring. The following are equivalent for any right R-module M and an integer n ≥ 0:

(1) afd(M) ≤ n.

(2) Torn+1(M, R/I) = 0 for any left annihilator I.

(3) Torn+j(M, R/I) = 0 for any left annihilator I and any j ≥ 1.

(4) There exists an exact sequence 0 → Fn → · · · → F1 → F0 → M → 0, where each Fi is A-flat.

PROOF : (1) ⇒ (2). Suppose afd(M) = m ≤ n. Let I be a left annihilator. Since R is a left ∼ ∼ AC ring, there exist left annihilators I1,I2, ··· ,In−m (write I0 = I) such that I = R/I1,I1 = ∼ ∼ ∼ ∼ R/I2, ··· ,In−m = R/In−m+1. Thus Torn+1(M, R/I) = Torn(M,I) = Torn(M, R/I1) = ∼ ∼ Torn−1(M,I1) = ··· = Torm+1(M, R/In−m) = 0.

(2) ⇒ (3) is easy by induction.

(3) ⇒ (4). There exists an exact sequence 0 → L → Fn−1 → Fn−2 → · · · → F0 → M → 0, where each Fi is projective. Then L is A-flat by dimension shifting.

(4) ⇒ (1) is obvious by dimension shifting.

Theorem 5.6 — The following are equivalent for a left AC ring R and an integer n ≥ 0 :

(1) l.ai − D(R) ≤ n.

(2) r.af − D(R) ≤ n.

(3) Extn+1(R/I, M) = 0 for any left annihilator I and any left R-module M.

(4) Torn+1(N, R/I) = 0 for any left annihilator I and any right R-module N.

PROOF : (1) ⇔ (3) follows from Proposition 5.4.

(2) ⇔ (4) holds by Proposition 5.5.

(3) ⇒ (4) follows from the isomorphism: 138 LIXIN MAO

n+1 + ∼ + Ext (R/I, N ) = Torn+1(N, R/I) for any left annihilator I and any right R-module N.

(4) ⇒ (3). Let I be any left annihilator and M any left R-module. Since R is a left AC ring, + n+1 + there is an exact sequence Torn+1(M , R/I) → (Ext (R/I, M)) → 0 by [5, Lemma 2.7 (1)], and so (3) holds. ¤

Corollary 5.7 — The following are equivalent for a left AC ring R:

(1) r.af − D(R) = 0.

(2) R is a Baer ring.

(3) Every right R-module is A-flat.

(4) Every cyclic torsionless left R-module is flat.

Remark 5.8 : (1) By Corollary 5.7, r.af − D(R) measures how far away a left AC ring R is from being a Baer ring.

(2) The condition that “R is a left AC ring” in Theorem 5.6 and Corollary 5.7 is not superfluous. In fact, assume that F is a field, and A = F × F × · · · . Let R be the subring of A consisting of

“sequences” (a1, a2, ···) ∈ A that are eventually constant. Then R is a commutative von Neumann regular ring (and hence r.af − D(R) = 0) but it is not an AC ring (and hence not a Baer ring) (see [18, Example 7.54]). This example shows that the inequality r.af − D(R) ≤ l.ai − D(R) may be strict. However, it is true that l.ai − D(R) = r.af − D(R) for any left AC ring R by Theorem 5.6.

Theorem 5.9 — The following are equivalent for a left AC ring R:

(1) l.ai − D(R) ≤ 1.

(2) r.af − D(R) ≤ 1.

(3) Every quotient module of any A-injective left R-module is A-injective.

(4) Every submodule of any A-flat right R-module is A-flat.

(5) Every left R-module has a monic AI-cover.

(6) Every right R-module has an epic AF-envelope.

PROOF : (1) ⇔ (2) follows from Theorem 5.6.

(1) ⇔ (3) holds by Proposition 5.4.

(2) ⇔ (4) follows from Proposition 5.5.

(4) ⇒ (6). For any right R-module M, there is an AF-preenvelope f : M → F by Theorem 4.7. Note that im(f) is A-flat by (4), so M → im(f) is an epic AF-preenvelope, and hence an epic AF-envelope.

(6) ⇒ (4). Let A be any submodule of an A-flat right R-module B. Since A has an epic RINGS CLOSE TO BAER 139

AF-envelope by (6), A has an isomorphic AF-envelope and so A is A-flat. (3) ⇒ (5). For any left R-module M, there is an AI-cover f : E → M by Theorem 4.7. Note that im(f) is A-injective by (3), so im(f) → M is a monic AI-precover, and hence a monic AI-cover.

(5) ⇒ (3). Let X be any A-injective left R-module and N any submodule of X. Since X/N has a monic AI-cover, X/N has an isomorphic AI-cover, and hence X/N is A-injective. ¤

Recall that φ : M → C is said to be a C-envelope with the unique mapping property [6] if for 0 0 0 any homomorphism f: M → C with C ∈ C, there is a unique homomorphism g : C → C such that gφ = f. Dually we have the definition of a C-cover with the unique mapping property.

Theorem 5.10 — The following are equivalent for a left AC ring R:

(1) l.ai − D(R) ≤ 2.

(2) r.af − D(R) ≤ 2.

(3) Every left R-module has an AI-cover with the unique mapping property.

Moreover, the above conditions hold if

(4) Every right R-module has an AF-envelope with the unique mapping property.

PROOF : (1) ⇔ (2) holds by Theorem 5.6.

(1) ⇒ (3). Let M be any left R-module. Then M has an AI-cover f : F → M by Theorem 4.7. It is enough to show that, for any G ∈ AI and any homomorphism g : G → F such that fg = 0, we have g = 0. In fact, there exists β : F/im(g) → M such that βπ = f since im(g) ⊆ ker(f), where π : F → F/im(g) is the natural map. Note that F/im(g) is A-injective by Proposition 5.4 since aid(ker(g)) ≤ 2. Thus there exists α : F/im(g) → F such that β = fα, and so fαπ = f. Hence απ is an isomorphism since f is a cover. Therefore π is monic, and so g = 0.

(3) ⇒ (1). Let M be any left R-module. Then we have the exact sequence

ϕ ψ 0 −→ M −→ E0−→E1 −→ N −→ 0 where E0 and E1 are injective. Let θ : H → N be an AI-cover with the unique mapping property. Then there exists δ : E1 → H such that ψ = θδ. Thus θδϕ = ψϕ = 0 = θ0, and hence δϕ = 0, which implies that ker(ψ) = im(ϕ) ⊆ ker(δ). Therefore there exists γ : N → H such that

γψ = δ, and so θγψ = ψ. Thus θγ = 1N since ψ is epic. It follows that N is isomorphic to a direct summand of H, and hence N is A-injective. So aid(M) ≤ 2 by Proposition 5.4, and hence l.ai − D(R) ≤ 2.

(4) ⇒ (2). Let M be any right R-module. Then we have the exact sequence

ψ ϕ 0 −→ N−→F1 −→ F0 −→ M −→ 0, 140 LIXIN MAO

where F0 and F1 are projective. Let θ : N → H be the AF-envelope with the unique mapping property. Then there exists δ : H → F1 such that ψ = δθ. Thus ϕδθ = ϕψ = 0, and hence ϕδ = 0, which implies that im(δ) ⊆ ker(ϕ) = im(ψ). So there exists γ : H → N such that

ψγ = δ, and hence ψγθ = ψ. Thus γθ = 1N since ψ is monic. Consequently N is isomorphic to a direct summand of H, and hence N is A-flat. Therefore afd(M) ≤ 2 by Proposition 5.5, and so r.af − D(R) ≤ 2. ¤

Finally, we give a class of left AC rings satisfying that l.ai − D(R) = 0 or ∞.

Lemma 5.11 — [12, Corollary 3.1.12] Let F be a class of modules closed under direct sums, extensions, continuous well ordered unions and contain all projective modules. If F ⊥ = S⊥ for a set S ⊆ F, then (F, F ⊥) is a cotorsion theory.

Theorem 5.12 — The following are equivalent for a left AC ring R:

(1) R is a left A-injective ring.

(2) Every right R-module has a monic AF-preenvelope.

(3) Every left R-module has an epic AI-cover.

(4) Every injective right R-module is A-flat.

(5) Every flat left R-module is A-injective.

If any of the above conditions holds, then l.ai − D(R) = r.af − D(R) = 0 or ∞.

PROOF : (1) ⇒ (2). Let M be any right R-module. Then M has an AF-preenvelope f : M → + F by Theorem 4.7. Since there is an exact sequence 0 → M → Π(RR) , M embeds in an A-flat right R-module by (1) and Theorem 4.5 (3). Thus f is a monomorphism.

(2) ⇒ (4) is obvious.

(4) ⇒ (5). Let M be a flat left R-module. Then M + is injective, and so M + is A-flat by (4). Thus M is A-injective by Theorem 4.5 (5).

(5) ⇒ (3). Let Card(R) = ℵβ and F ∈ AI. By [9, Lemma 5.3.12], for each x ∈ F , there is a pure submodule S of F with x ∈ S such that Card(S) ≤ ℵβ (simply let N = Rx and f = 1N in the lemma). So we can write F as a union of a continuous chain (Fα)α<λ of pure submodules of F such that Card(F0) ≤ ℵβ and Card(Fα+1/Fα) ≤ ℵβ whenever α + 1 < λ. If N is a left 1 1 R-module such that Ext (F0,N) = 0 and Ext (Fα+1/Fα,N) = 0 whenever α + 1 < λ, then 1 Ext (F,N) = 0 by [7, Lemma 1] or [9, Theorem 7.3.4]. Since Fα is a pure submodule of F + + + + for any α < λ, F → Fα → 0 is split. Then Fα ∈ AF since F ∈ AF by Theorem 4.5, and so Fα ∈ AI. On the other hand, Fα is a pure submodule of Fα+1 whenever α + 1 < λ, so the exact sequence 0 → Fα → Fα+1 → Fα+1/Fα → 0 induces the split exact sequence + + + + + 0 → (Fα+1/Fα) → Fα+1 → Fα → 0. Thus (Fα+1/Fα) ∈ AF since Fα+1 ∈ AF, and hence Fα+1/Fα ∈ AI. Let X be a set of representatives of all modules G ∈ AI with Card(G) ≤ ℵβ. RINGS CLOSE TO BAER 141

Then AI⊥ = X⊥.

We note that AI is closed under direct sums, extensions, direct limits (in particular, continuous well ordered unions) by Proposition 3.3 and Theorem 4.5 since R is a left AC ring, and AI contains all projective left modules by (5). Therefore (AI, AI⊥) is a cotorsion theory by Lemma 5.11.

Since (AI, AI⊥) is cogenerated by the set X,(AI, AI⊥) is a complete cotorsion theory by [7, Theorem 10]. Therefore every left R-module has an epic AI-cover by [9, Theorem 7.2.6] (for AI is closed under direct limits).

(3) ⇒ (1) is clear.

Now suppose that any of the equivalent conditions above holds, and l.ai − D(R) = n < ∞.

For any left R-module M, there exists an exact sequence 0 → L → Fn−1 → Fn−2 → · · · →

F0 → M → 0, where each Fi is projective. Thus each Fi is A-injective. So M is A-injective since aid(L) ≤ n. It follows that l.ai − D(R) = 0. ¤

Remark 5.13 : Let R = Z4. Then R is a commutative QF ring. So l.ai − D(R) = r.af − D(R) = ∞ by Theorem 5.12 since R is not a Baer ring.

ACKNOWLEDGEMENT

This research was partially supported by SRFDP (20050284015), China Postdoctoral Science Foun- dation (20060390926), Collegial Natural Science Research Program of Education Department of Jiangsu Province (06KJB110033), and Jiangsu Planned Projects for Postdoctoral Research Funds (0601021B). The author would like to thank Professor Nanqing Ding for his useful advices and the referee for the valuable comments and suggestions in shaping the paper into its present form.

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