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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-Injective Module; 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 Á : M ! C is a C-preenvelope if C 2 C and the abelian group homomorphism HomR(Á; C ): 0 0 0 HomR(C; C ) ! HomR(M; C ) is an epimorphism for each C 2 C.A C-preenvelope Á : M ! C is said to be a C-envelope if every endomorphism g : C ! C such that gÁ = Á 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 = fC : ExtR(L; C) = 0 for all L 2 Lg ? 1 the right orthogonal class of L, and by L = fC : ExtR(C; L) = 0 for all L 2 Lg 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 2 C and F 2 F, and 0 ! K ! G ! M ! 0 with G 2 F and K 2 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 2 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 2 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 2 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 idempotent 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 functional analysis. 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 1. 3. A-INJECTIVE MODULES AND 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.
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