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Modules, lattices and their direct summands

It is well known that any finitely generated Zmodule is a direct sum of a projective (in fact a free) module and a Noetherian module (in fact a module of finite length) (for example see [Fu]). More generally, [Sm1] proved that if R is a right Noetherian ring with maximal Artinian right ideal A, then every finitely generated right Rmodule is the direct sum of a projective module and a module of finite length if and only if the ideal A = eR for some idempotent e in R and the ring R/A is a left and right hereditary left and right Noetherian semiprime ring (see [Sm1, Theorem 3.3]). It was left open in [Sm1] whether the assumption that R be right Noetherian is necessary. In fact, it is not, as Chatters [Ch] showed, by proving that if R is a ring such that every cyclic right Rmodule is the direct sum of a projective module and a Noetherian module, then R is a right Noetherian ring (see [Ch, Theorem 3.1]). Chatters [Ch, Theorem 4.1] also proved that if a is an ordinal and R a ring such that every cyclic right Rmodule is the direct sum of a projective module and a module of Krull dimension at most a, then the right Rmodule R has Krull dimension at most ? + 1. Van Huynh and Dan [HD] have considered rings with the property that every cyclic right module is the direct sum of a projective module and an Artinian module, or the property that every cyclic right module is the direct sum of a projective module and a semisimple module. This led to the investigations in [SHD] and [Sm2]. The following terminology was introduced. Let X be a class of modules. Then hX is defined to be the class of modules M such that for each submodule N of H, M/N belongs to X. Moreover, dX is defined to be the class of modules M such that for each submodule N of M, there exists a direct summand K of M such that N ? K and K/N belongs to X. Finally, eX is defined to be the class of modules M such that for each essential submodule E of M, M/E belongs to X. It is proved in [Sm2] that when X is the class U: the class of modules with finite uniform dimension, then a module M belongs to eU if and only if M/N belongs to hU for some semisimple submodule N of M (Theorem 1.2.1). This fact led [Sm2] to prove that a module M belongs to dU if and only if M = M1 ⊕ M2 where is a semisimple module and M2 belongs to hU (Theorem 1.2.3). Moreover, [Sm2] proved that when X is the class N; the class of Noetherian modules, or when X is the class K: the class of modules which have Krull dimension then a module M belongs to dN (respectively, dK) if and only if M = M1 ⊕ M2 where M1 is semisimple and M2 belongs to N (respectively, K) (Theorem 1.2.4). In the first two sections of chapter I of this thesis, we present all of the background material from [SHD] and [Sm2] and, for completeness, we include the proofs. In the third section, we prove a generalization of Theorem 1.2.4. i.e. we prove that when X is the class of modules with dual Krull dimension at most alpha, for some ordinal alpha > 0, then a module M belongs to dX if and only if M = M1 ⊕ M2 where M, is a semisimple module and M2 belongs to X (Theorem 1.3.11). In section 2.1, we define the property (P) : a module M satisfies (P) provided that for any submodule N of M, there exists a direct summand K of M such that Soc K ⊆ N ⊆ K. We prove that a module M is the direct sum of modules with (P) and M is eventually semisimple if and only if M = M1 ⊕ M2 ⊕ M3 where M1 is a semisimple module, M2 a finite direct sum of uniform modules and M3 has finite uniform dimension and zero socle (Theorem 2.1.5). In section 2.2, we define the property (P*): a module M satisfies (P*) provided that for any submodule N of M, there exists a direct summand K of M with K ⊆ N and N/K ⊆ Rad M/K. We prove that a module M is a direct sum of modules satisfying (P*) and the radical of M has finite uniform dimension if and only if M= M1 ⊕ M2 ⊕ M3 where M, is a semisimple module M2 is a radical module with finite uniform dimension and M3 is a finite direct sum of local submodules and has finite uniform dimension (Theorem 2.2.8). In chapter III, we define h*X (respectively, e*X) to be the class of modules M such that every (small) submodule of M belongs to X. Moreover, we define d*X to be the class of modules M such that for each submodule N of M, N contains a direct siimmand K of M such that N/K belongs to X.
