Distributive and Multiplication Modules and Rings

We study rings in which every ideal is a finitely generated multiplication right ideal.     

Distributive extensions of modules

Let X be a submodule of a module M. The extension is said to be distributive if X ∩ (Y + Z) = X ∩ Y + X ∩ Z for any two submodules Y and Z of M. We study distributive extensions of modules over not necessarily commutative rings. In particular, it is proved that the following three conditions are equivalent: (1) is a distributive extension; (2) for any submodule Y of the module M, no simple subfactor of the module X/(X∩Y ) is isomorphic to any simple subfactor of Y/(X∩Y) (3) for any two elements x ∈ X and m ∈ M, there does not exist a simple factor module of the cyclic module xA/(X ∩ mA) that is isomorphic to a simple factor module of the cyclic module mA/(X ∩ mA). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 141–150, 2006.     

Modules with Nakayama’s property

Modules M _A with Nakayama’s property are studied. In particular, for a right invariant ring A, it is proved that all right A-modules satisfy Nakayama’s property if and only if the ring A is right perfect.     

Rings over which all modules are I 0-modules. II

All right R-modules are I ₀-modules if and only if either R is a right SV-ring or R/I ⁽²⁾ (R) is an Artinian serial ring such that the square of the Jacobson radical of R/I ⁽²⁾ (R) is equal to zero.     

Bezout modules and rings

For any ring A, there exist a Bezout ring R and an idempotent e ∈ R with A ? eRe. Every module over any ring is a direct summand of an endo-Bezout module. Over any ring, every free module of infinite rank is an endo-Bezout module.     

Distributive skew Laurent polynomial rings

We describe skew Laurent polynomial rings that are right distributive.