Pure submodule

Abstract: Introduction. In this paper several properties of absolutely pure modules are given. It is shown that absolutely pure and injective are equivalent properties for modules over Dedekind rings. However, it is proved that absolutely pure and injective are not equivalent properties for modules over rings which are not Noetherian. That every module has a maximal absolutely pure submodule is also established. A sufficient condition for the uniqueness of a maximal absolutely pure submodule is also given. This paper constitutes a portion of the author's doctoral dissertation written at the University of South Carolina where he held a Cooperative Graduate Fellowship. The writer is indebted to Professor Edgar Enochs who suggested this topic and directed its development while providing sufficient inspiration and assistance and, most of all, exhibiting infinite patience. In this paper all rings will have a unit and all modules will be unitary. A will always denote a ring. We agree that if E' is a submodule of E and v. E'—>E is the canonical injection then the map 1 ®v: F®E' —>P £ will be called the canonical map where 1: F-^F is the identity map of P. If the canonical map is an injection for all P, then E' is said to be a pure submodule of P. Observe that if E' is a pure submodule of E then aEC\E' =aE' for all nonzero aEA by examining the diagram

Abstract: We characterize left coherent, left Noetherian, and left Artinian rings by the flatness, projectivity, and injectivity of the character module of certain left R-modules. Introduction. The character module of a left (right) R-module M is the right (left) R-module M+ = Homz(M, Q/Z) where Z denotes the group of integers and Q denotes the additive group of rational numbers. Character modules have played an important role in the study of rings through their modules. See, for example, [3]-[6], [8], and [9]. Lambek [6] proved that, over any ring, a module is flat if and only if its character module is injective (equivalently absolutely pure). It is easy to show that a left perfect ring is characterized by the property: A module is projective if and only if its character module is injective. We shall consider the dual conditions and determine those classes of rings for which a module has one of the properties absolutely pure or injective if and only if its character module has one of the dual properties flat or projective. The main result of this paper is a characterization of a left Artinian ring as a ring over which a left R-module is injective if and only if its character module is projective. This removes an unnecessary hypothesis from Ramamurthi [8, Proposition 6, p. 182] and proves the converse. Results. Let R denote an associative ring with identity. Following Ramamurthi [8] we call an R-module M a PC-module (FC-module) if M + is a projective (respectively flat) R-module on the opposite side. A module is absolutely pure if it is a pure submodule of every over-module. The following lemma will be useful. LEMMA 1. Let I be any index set. Let (A,)i , be any family of left (or right) R-modules. Then (1) e Ai is a pure submodule of II Ai. (2) If for each i e I, Bi is a pure submodule of Ai, then II Bi is a pure submodule of IAi. The proof of Lemma 1 is straightforward. We note that if M is an FC-module then M is absolutely pure. This is an easy consequence of the fact that every character module is pure-injective. We begin by studying the weakest of the four conditions mentioned in the introduction. A result of Wurfel [9] is included for completeness. Received by the editors April 1, 1980. 1980 Mathematics Subject Classification. Primary 16A50, 16A52.

Abstract: As is well-known, the infinite module theory of certain types of finite-dimensional algebras has a strong similarity to abelian group theory. (See [1] for the case of Kronecker modules, [28] for the representations of tame hereditary algebras). The same phenomenon occurs for k-linear representations of (suitably restricted) ordered sets [18, 19]. In all these cases there is a primary decomposition for torsion-modules (cf. Cor. 2.6), the torsion submodule is always a pure submodule (cf. Prop. 3.1) and torsion-free modules behave in some respect like flat modules (cf. Prop. 2.4). Further, divisible modules are algebraically compact (cf. Thms. 4.4 and 4.6), in some cases even injective [19].

Abstract: Let A be an integral domain and K its field of fractions. An Amodule E is said to be torsion free if ax = 0 for azA, x EE implies a = 0 or x = 0. We will say that a submodule E1 of an A -module E is pure in E if axE,=acEC\E1 for all aGA. Then if E is torsion free, a submodule E1 of E is pure in E if and only if E/E, is torsion free. Clearly the union of a chain of pure submodules of a module is still a pure submodule and if E2CE1, are submodules of E such that E2 is pure in E1 and E1/E2 pure in E/E2 then El is pure in E. It is well known that for any A-module E there exists a torsion free A -module E1 and an epimorphism p: E-*E1 such that if 4 is any linear mapping from E into a torsion free module F then there is a unique linear mapping f: El-*F such that f o p =4, i.e., the diagram