Supplements in Coatomic Modules Having the Complete Max-Property

Abstract

Let R be a ring with identity. A right R-module M has the complete max-property if the maximal submodules of M are completely coindependent (i.e., every maximal submodule of M does not contain the intersection of the other maximal submodules of M ). A right R-module is said to be a good module provided every proper submodule of M containing Rad(M ) is an intersection of maximal submodules of M . We obtain a new characterization of good modules. Also, we study good modules which have the complete maxproperty. The second part of this paper is devoted to investigate supplements in a coatomic module which has the complete max-property. Mathematics Subject Classification (2010): 16D10, 16D99 Let R be a unitary ring and M a right R-module. A submodule N of M is called small in M (written N M ) if for every proper submodule L of M , N + L = M . A submodule L of M is called coclosed in M if L/K is not small in M/K for any proper submodule K of L. We denote by Rad(M ) the radical of M . A module M is called coatomic if every proper submodule of M is contained in a maximal submodule, that is, Rad(M/N ) = 0 for every proper submodule N ≤ M . Let L be a submodule of M . A submodule K of M is called a supplement of L in M if K is minimal with respect to the property M = L + K; equivalently, M = L + Kand K ∩ L K. A submodule P of M is called a supplement submodule if P is a supplement of some submodule of M . The module M is called supplemented if every submodule of M has a supplement in M . A module M is called semilocal if M/ Rad(M ) is semisimple. A module M is called cosemisimple (or a V-module) if every simple R-module is M -injective, or equivalently, every proper submodule of M is an intersection of maximal submodules (see [7, 23.1]). A module M is called a SUPPLEMENTS IN COATOMIC MODULES 19 good module if M/ Rad(M ) is a cosemisimple module (see [7, 23.3]). A non-empty family of submodules N i (i ∈ I) of a module M is called coindependent if, for any j ∈ I and any finite subset J of I \ {j}, N j + i∈J N i = M . The family N i (i ∈ I) is called completely coindependent if, for every j ∈ I, N j + i =j N i = M (see [4, p. 8]). Following [6, p. 74], a module M is said to have the complete max-property if the maximal submodules of M form a completely coindependent set of submodules of M . In this paper, we adopt the convention that the intersection of an empty set of submodules of a module M is M itself.In Section 2, we provide some new characterizations of good modules (Theorem 2.3). Also, we investigate the interplay between the complete max-property and each one of the properties coatomic and good.The investigations in Section 3 focus on supplements in a coatomic module which has the complete max-property. After characterizing them, we show that for a coatomic module M , if M has the complete max-property, then any supplement submodule in M has also the complete max-property. In addition, we prove that if M is a coatomic module which has the complete max-property and F is a supple-Throughout this paper, R will denote an ...