Journal of Mathematical Sciences, Vol. 164, No. 1, 2010 SUBMODULES AND DIRECT SUMMANDS A. N. Abyzov and A. A. Tuganbaev UDC 512.55 Abstract. This paper contains new and known results on modules in which submodules are close to direct summands. The main results are presented with proofs. All rings are assumed to be associative and with nonzero identity element. Expressions such as “an Artinian ring” mean that the corresponding right and left conditions hold. For a module M , a sub- module X in M is said to be small in M if X + P = M for every proper submodule P of M . We call a module M an I 0 module (cf. [30]) if every finitely generated nonsmall submodule of M contains a nonzero direct summand of M . A module M is said to be weakly regular if every submodule of M that is not contained in the Jacobson radical of M , contains a nonzero direct summand of M . Weakly regular modules coincide with I 0 modules; they are studied in [1–5, 24, 28, 30, 43–46], [42, Chap. 3], and other papers. A ring R is called a right generalized SV-ring if every right R-module is weakly regular. For a module M , the intersection of all maximal submodules of M is denoted by J (M ); it is called the Jacobson radical of M . For a module M , the injective hull of M is denoted by E(M ). A ring A is said to be regular (in the sense of von Neumann) if a ∈ aAa for every element a ∈ A. A module M is said to be uniserial if any two submodules of M are comparable with respect to inclusion. A direct sum of uniserial modules is called a serial module. A module M is said to be semisimple if every submodule of M is a direct summand in M . For a module M , a submodule N of M is said to be essential if X ∩ N = 0 for every nonzero submodule X of M . In this case, M is also called an essential extension of the module N . A module M is said to be uniform if any two nonzero submodules in M have nonzero intersection. A module M is said to be semi-Artinian if every factor module of M is an essential extension of a semisimple module. If M is a module, then any module that is isomorphic to a submodule of some homomorphic image of a direct sum of copies of M is called an M -subgenerated module. The full subcategory of all right R-modules that consists of all M -subgenerated modules is denoted by σ(M ) and is called the Wisbauer category of the module M . For a module M and a submodule N in M , we say that N lies over a direct summand of M if there exist submodules N 1 and N 2 such that N 1 ⊕ N 2 = M , N 1 ⊂ N , and N 2 ∩ N is small in N 2 . A right R-module M is called a lifting module if every submodule of M lies over a direct summand of M . If M is a module and every cyclic submodule of M lies over a direct summand of M , then the module M is said to be semiregular. A module M is said to be a CS module if every submodule of M is an essential submodule in some direct summand of M . For a module M , the Loewy series of M is the ascending chain 0 ⊂ Soc 1 (M ) = Soc(M ) ⊂···⊂ Soc α (M ) ⊂ Soc α+1 (M ) ⊂ ..., where Soc α (M )/ Soc α−1 (M ) = Soc ( M/ Soc α−1 (M ) ) for every nonlimit ordinal number α and Soc α (M )=  β<α Soc β (M ) for every limit ordinal number α. We denote by L(M ) the submodule of the form Soc ξ (M ), where ξ is the the least ordinal number with Soc ξ (M ) = Soc ξ+1 (M ). A module M is semi-Artinian if and only if Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 14, No. 6, pp. 3–31, 2008. 1072–3374/10/1641–0001 c  2010 Springer Science+Business Media, Inc. 1