Journal of Mathematical Sciences, "Col. 93, No. 2, 1999 DISTRIBUTIVE MODULES AND RINGS AND THEIR CLOSE ANALOGS A. V. Mikhalev and A. A. Tuganbaev UDC .512.55 Introduction A module M is called distributive if the lattice Lat(M) of all its submodules is distributive, i.e., FIq(G+H) = FnG+Ff3H for all submodules F, G, and H of the module M. A module M is called uniserial if any of its submodules is comparable with respect to inclusion, i.e., the lattice Lat(M) is a chain. The class of distributive modules contains properly all uniserial modules, and in particular, all simple modules. All strongly regular rings (for example, all quotient rings of direct products of skew fields and all com- mutative regular rings) are distributive. All commutative Dedekind rings (for example, the rings of integer algebraic numbers or commutative principal ideal rings) and all valuation rings in skew fields are also dis- tributive. We shall call a module M a Bezout or locally cyclic module if any of its finitely generated submodules is cyclic. If all maximal right ideals of a ring A are ideals (for example, if A is commutative), then all Bezout A-modules axe distributive (see 2.12). The rings each of whose finitely generated modules is decomposed into a direct sum of distributive modules (for example, all Artinian serial and Dedekind rings are Of this type) form a sufficiently wide class. If A is a commutative regular ring (for example, a field) and G is a locally cyclic group, then the group ring A[G] and the polynomial rings A[z] and A[z, z -1] are distributive Bezout rings. We note [81, 76, 119, 120, 77] among the first works concerning distributive modules and rings in the noncommutative case. The systematic study of distributive modules over noncommutative rings has its origin in [82, 71, 125, 92, 83]. Distributive modules were considered in [78, Chap. 9; 15, Sec. 4.1; 19, Sec. 2.2]. In [7], distributive modules are applied to complex analysis. In [6], distributive rings axe used for the study of rings of continuous functions in topological spaces. In [130], distributive modules are applied to a study of rings with a duality. In [113, 31, 37, 38], the distributivity is applied to rings of weak global dimension one and to hereditary rings. In [35, 38, 44], a number of applications of distributive rings and modules to formal power series rings were obtained. Distributive group and semigroup rings were studied in [14, 111, 95, 34, 36, 38, 39, 41, 51]. Distributive quaternion algebras were studied in [52-54]. In [32, 39, 20, 57] modules which are distributive over their rings of endomorphisms were studied. Topological aspects of properties of distributive modules and rings were considered in [127, 5, 39]. Distributive modules over incidence algebras were studied in [98]. Modules decomposed into a direct sum of distributive modules and rings that are a direct sum of distributive right ideals were studied in [100, 101, 22, 10-12, 91, 80, 130]. The results concerning distributive modules and rings were addressed in surveys [17, 18, 3]. In this paper, as a rule, we consider distributive modules over noncommutative rings. Since the distribu- tivity of a noncommutative ring A is equivalent to the fact that all localizations of the ring A with respect to its maximal ideals are uniserial rings [113] (in particular, all Prfifer domains are distributive), the commutative case deserves special consideration. Here, we just mention [113, 114, 123, 75, 127, 50]. Moreover, it is worth Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 40, Algebra-6, 1996. 1072-3374/99/9302-0149522.00 9 Kluwer Academic/Plenum Publishers 149